Smooth Bezier Surfaces over Arbitrary Quadrilateral Meshes
SSmooth B´ezier Surfaces over UnstructuredQuadrilateral Meshes
Michel Bercovier ∗ and Tanya Matskewich The Rachel and Selim Benin School of Computer Science and Engineering ,Hebrew University of Jerusalem,Israel. Microsoft Corp. Redmond, WA, USA,
Abstract
We study the following problem: given a polynomial order of approximation n and thecorresponding B´ezier tensor product patches over an unstructured quadrilateral meshmade of convex quadrilaterals with vertices of any valence , is there a solution to the G ( and as a consequence the C ) approximation (resp. interpolation ) problem ?To illustrate the interpolation case , constraints defining regularity conditions acrosspatches have to be satisfied. The resulting number of free degrees of freedom mustbe such that the interpolation problem has a solution! This is similar to studying theminimal determining set (MDS) for a C continuity construction.Based on the equivalence of G and C we introduce a sufficient G conditionthat is better adapted to the present problem. Boundary conditions are then analysedincluding normal derivative constraints ( common in FEM but not in CAGD. )The MDS are constructed for both polygonal meshes and meshes with G -smoothpiecewise B´ezier cubic global boundary. The main results are that such MDS existsalways for patches of order ≥
5. For n = 4 criterions for mesh structures avoidingunder constrained situations are analysed . This leads to the construction of bases bysolving a well defined linear system, which allows the solution of the problem for largefamilies structures of planar meshes, without using macro-elements or subdivisions. .A complete solution for cubic C boundaries is given , again by a constructivealgorithm. We also show that one can mixes quartic and quintic patches. constraints.Explicit construction is provided for important types of interpolation/boundary Fi-nally some numerical examples are given . As a conclusion from a practical pointof view, the present paper provides a way to solve C interpolations/approximationsand fourth order partial differential problems on arbitrary structures of quadrilateralmeshes. Defining a global in-plane parametrisation a priory allows the introduction ofa ”physical” energy functional, as is done in Isogeometric Analysis, energy that relatesto the functional representation of the surface. ∗ E-Mail : [email protected] ;Corresponding author i a r X i v : . [ m a t h . NA ] J a n i Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
I Introduction
11 Problem definition , current sate and aim of the research . . . . 11.1 Description of the problem . . . . . . . . . . . . . . . . . . . . . . 11.2 Brief review of related works . . . . . . . . . . . . . . . . . . . . 11.2.1 CAGD based constructions . . . . . . . . . . . . . . . . . 21.2.2 Interaction between FEM and CAGD . . . . . . . . . . . 31.3 The principal aim of the present work . . . . . . . . . . . . . . . 51.3.1 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.2 Definitions and Hypotheses . . . . . . . . . . . . . . . . . 61.4 Domains of application . . . . . . . . . . . . . . . . . . . . . . . . 81.4.1 Solution of an interpolation/approximation problem usingthe functional form of the shape functional . . . . . . . . 81.4.2 Approximate solution of a partial differential equationover an arbitrary quadrilateral mesh . . . . . . . . . . . . 91.5 Structure of the work . . . . . . . . . . . . . . . . . . . . . . . . 92 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1 Points and vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Planar mesh data . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.1 Vertices, edges and twist characteristics . . . . . . . . . . 122.4 Partial derivatives of in-plane parametrisations . . . . . . . . . . 132.5 Weight functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.6 3 D data of the resulting surface . . . . . . . . . . . . . . . . . . . 142.6.1 B´ezier control points of two adjacent patches . . . . . . . 142.6.2 B´ezier control points adjacent to some mesh vertex . . . . 142.6.3 Partial derivatives of patches at the common vertex . . . 142.7 Definitions of special sets, spaces and equations . . . . . . . . . . 15 II Some fundamental results regarding G -smooth surfaces
213 Basic definitions related to smoothness of the surface . . . . . . . 214 The vertex enclosure problem . . . . . . . . . . . . . . . . . . . . 224.1 General formulation of the vertex enclosure constraint . . . . . . 22 ii III General linearisation method
265 Linearisation of the minimisation problem . . . . . . . . . . . . . 265.1 Linearisation of the smoothness condition . . . . . . . . . . . . . 265.2 Linear form of ”additional” constraints . . . . . . . . . . . . . . . 285.2.1 Interpolation constraints . . . . . . . . . . . . . . . . . . . 285.2.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . 285.3 Quadratic form of the energy functional . . . . . . . . . . . . . . 296 Principles of construction of MDS . . . . . . . . . . . . . . . . . 316.1 Special subsets of control points and their dimensionality . . . . 316.2 Relation between MDS and the ”additional” constraints . . . . . 316.3 Principle of locality in construction of MDS . . . . . . . . . . . . 326.4 Aim of the classification process . . . . . . . . . . . . . . . . . . . 327 From MDS to solution of the linear minimisation problem . . . . 33
IV MDS for a quadrilateral mesh with a polygonal global boundary
358 Mesh limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 In-plane parametrisation . . . . . . . . . . . . . . . . . . . . . . . 3610 Conventional weight functions and linear form of G -continuityconditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3611 Local MDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3711.1 Local classification of E , V -type control points for a separate ver-tex based on ” Eq (0)”-type equations . . . . . . . . . . . . . . . . 3811.1.1 Formal equation and geometrical formulation . . . . . . . 3811.1.2 Degrees of freedom and dependencies . . . . . . . . . . . . 3811.1.3 Local templates for a separate inner vertex . . . . . . . . 3911.1.4 Local templates for a separate boundary vertex . . . . . . 3911.2 Local classification of D , T -type control points for a separate ver-tex based on ” Eq (1)”-type equations . . . . . . . . . . . . . . . . 4011.2.1 Formal equation and geometrical formulation . . . . . . . 4011.2.2 Theoretical results for an inner vertex . . . . . . . . . . . 4111.2.3 Local templates for a separate inner vertex . . . . . . . . 4311.2.4 Local templates for a separate boundary vertex . . . . . . 4411.3 Local classification of the middle control points for a separate edge 4511.3.1 Existence and types of the local templates . . . . . . . . . 4611.3.2 Example of the local MDS for n = 4 and n = 5 . . . . . . 4712 Global MDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5012.1 MDS of degree n ≥ n = 4 . . . . . . . . . . . . . . . . . . . . . . . . . 51 v D -type control points . . 5112.2.2 Examples of possible difficulties . . . . . . . . . . . . . . . 5212.2.3 Sufficient conditions and algorithms for the global classi-fication of D , T -type control points . . . . . . . . . . . . . 5212.2.4 The existence of MDS. Analysis of different ”additional”constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . 5712.3 Dimensionality of MDS . . . . . . . . . . . . . . . . . . . . . . . 57 V MDS for a quadrilateral mesh with a smooth global boundary ( bicubic ) . . . . . . . . . . . . . 6514 Conventional weight functions . . . . . . . . . . . . . . . . . . . . 6714.1 Weight functions for an edge with two inner vertices . . . . . . . 6714.2 Weight functions for an edge with one boundary vertex . . . . . 6714.2.1 Partial derivatives of in-plane parametrisations along theedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6714.2.2 Weight functions . . . . . . . . . . . . . . . . . . . . . . . 6814.2.3 Geometrical meaning of the actual degrees of the weightfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 6915 Linear form of G -continuity conditions . . . . . . . . . . . . . . 7115.1 G -continuity conditions for an edge with two inner vertices . . . 7115.2 G -continuity conditions for an edge with one boundary vertex . 7215.2.1 Formal construction of the linear system of equations . . 7215.2.2 An equivalent system of the linear equations . . . . . . . 7315.2.3 Some important properties of ”sumC-equation” . . . . . . 7316 Local MDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7416.1 Local templates for a separate vertex . . . . . . . . . . . . . . . . 7416.1.1 Local templates for an inner vertex . . . . . . . . . . . . . 7416.1.2 Local templates for a boundary vertex . . . . . . . . . . . 7516.2 Local classification of the middle control points for a separate edge 7616.2.1 Local templates for an edge with two inner vertices . . . . 7616.2.2 Local templates for an edge with one boundary vertex . . 76
17 Global MDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8117.1 Algorithm for construction . . . . . . . . . . . . . . . . . . . . . . 8117.2 Existence of global MDS of degree n ≥ n = 4. Analysis of different”additional” constraints. . . . . . . . . . . . . . . . . . . . . . . . 8217.4 Dimensionality of MDS . . . . . . . . . . . . . . . . . . . . . . . 83 VI Mixed MDS of degrees and VII Computational examples
VIII Conclusions and further research. Bibliography Appendix
1A Illustrations to approximate solution of the Thin Plate Problem . 1B Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4C Technical Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . 8D Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15E Computation of Thin Plate energy for bilinear in-plane parametri-sation of mesh element . . . . . . . . . . . . . . . . . . . . . . . . 55F From MDS to solution of the quadratic minimisation problem . . 58G Construction of an interpolating surface: current approach andinterpolation based on 3 D mesh of curves . . . . . . . . . . . . . 60 i List of Figures D B´ezier parametric patch and the corresponding planar meshelement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Planar mesh and 3 D surface interpolating 3 D points and tangentplanes at the mesh vertices. . . . . . . . . . . . . . . . . . . . . . 114 A planar mesh element. . . . . . . . . . . . . . . . . . . . . . . . 175 Two adjacent planar mesh elements. . . . . . . . . . . . . . . . . 176 Vectors important for computation of the actual degrees of theweight functions in the case of the bilinear parametrisation of twoadjacent mesh elements. . . . . . . . . . . . . . . . . . . . . . . . 187 Planar mesh elements adjacent to the common vertex. . . . . . . 188 Partial derivatives of in-plane parametrisations for two adjacentpatches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 B´ezier control points adjacent to the common edge of two patches. 1910 B´ezier control points adjacent to some mesh vertex. . . . . . . . 2011 Two adjacent parametric patches with G -smooth concatenationalong the common boundary. . . . . . . . . . . . . . . . . . . . . 2012 Control points involved into the boundary conditions. . . . . . . 3013 Control points, which do not participate in G -continuity con-ditions (a) Inner element (b) Corner element (c) Boundary non-corner element. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3414 Control points of a global in-plane parametrisation. . . . . . . . . 3415 An illustration for the mesh limitations. . . . . . . . . . . . . . . 3516 Examples of meshes, which do not satisfy the ”Uniform EdgeDistribution Condition” . . . . . . . . . . . . . . . . . . . . . . . . 3617 Local templates for the classification of V , E -type control pointsin case of global bilinear in-plane parametrisation ˜Π ( bilinear ) . . . 4818 Possible mesh configuration, which automatically satisfies the ”Circular Constraint” . . . . . . . . . . . . . . . . . . . . . . . . . 4819 Local templates for the classification of D , T -type control pointsadjacent to an inner vertex in case of global bilinear in-planeparametrisation ˜Π ( bilinear ) . . . . . . . . . . . . . . . . . . . . . . 4820 Local templates for the classification of D , T -type control pointsadjacent to a boundary vertex in case of global bilinear in-planeparametrisation ˜Π ( bilinear ) . . . . . . . . . . . . . . . . . . . . . . 4921 An illustration for the ”Projections Relation” . . . . . . . . . . . . 4922 Local templates for the classification of the middle control pointsin case of global bilinear in-plane parametrisation ˜Π ( bilinear ) . . . 4923 An example of a ”pure” global MDS ˜ B (5) ( ˜Π ( bilinear ) . . . . . . . . 58 ii
24 An example of a global MDS ˜ B (4) ( ˜Π ( bilinear ) , which fits the clampedboundary conditions. . . . . . . . . . . . . . . . . . . . . . . . . . 5925 Examples of such meshes that not every D -relevant mesh vertexhas its own D -type control point. . . . . . . . . . . . . . . . . . . 6026 An example of mesh configuration when 3-vertex does not con-tribute to the ”Circular Constraint” for adjacent 4-vertices. . . . 6027 An illustrations for the construction of the D -dependency graph. 6028 Construction of D -dependency tree for a connected component of D -dependency graph. (a) Connected component of D -dependencygraph (here the structure of the component is not correct in themeaning that it does not correspond to any planar mesh, the Fig-ure serves only as an illustration for Algorithm 3). (b) Spanningtree of the connected component. (c) D -dependency tree of theconnected component. . . . . . . . . . . . . . . . . . . . . . . . . 6129 An example of two different D -dependency trees connected bythe directed edges at the root vertices. (a) Planar mesh verticesand edges. (b) D -dependency graph consisting of two connectedcomponents. (c) D -dependency trees for two components of D -dependency graph. . . . . . . . . . . . . . . . . . . . . . . . . . 6130 An example of the classification of D -type control points. . . . . 6231 Regularization of an inner 4-vertex. Here ˜ V ( init ) is a vertex of theinitial mesh and ˜ V ( reg ) is the corresponding regularized vertex. In D -dependency graph, ˜ V ( reg ) helps to obtain dangling half-edgesfor connected components of the adjacent vertices. . . . . . . . . 6232 An example of a ”pure” global MDS ˜ B (4) ( ˜Π ( bilinear ) . . . . . . . . 6233 An example of a planar domain with a smooth global boundary. 6334 A planar mesh element, control points of a boundary curve andcontrol points of in-plane parametrisation in the case of a meshwith a smooth global boundary. . . . . . . . . . . . . . . . . . . . 6635 Two adjacent boundary mesh elements in the case of a mesh witha smooth global boundary. . . . . . . . . . . . . . . . . . . . . . . 7036 Coefficients of ˜ L u , ˜ R u , ˜ L v with respect to B´ezier and to powerbases for two adjacent mesh elements in the case of global in-plane parametrisation ˜Π ( bicubic ) . . . . . . . . . . . . . . . . . . . . 7037 Different geometrical configurations of two adjacent boundarymesh elements lead to different actual degrees of the conventionalweight functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7138 Local templates for classification of V , E -type control points adja-cent to a boundary vertex in the case of global in-plane parametri-sation ˜Π ( bicubic ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8039 Local templates for an edge with one boundary vertex in the caseof global in-plane parametrisation ˜Π ( bicubic ) . . . . . . . . . . . . . 8040 An illustration for degree elevation for a boundary mesh element. 8641 Examples of ˜ B (4) for irregular 4-element mesh. . . . . . . . . . . 88 iii
42 An irregular quadrilateral mesh for a circular domain. . . . . . . 9243 The resulting smooth surface (case of the circular domain, irreg-ular mesh). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9244 Irregular quadrilateral mesh for a square domain. . . . . . . . . . 9345 The resulting smooth surface (case of the square domain, irregu-lar mesh). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Appendix
46 Level lines for the resulting surface and for its first-order deriva-tives (case of the circular domain, irregular mesh) (a) Z (b) ∂Z∂X (c) ∂Z∂Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Level lines for the resulting surface and for its first-order deriva-tives (case of the square domain, irregular mesh) (a) Z (b) ∂Z∂X (c) ∂Z∂Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Values for approximate (bold line) and exact (thin line) solutions,their first order derivatives and the bending moments along seg-ment X = 1 (case of the circular domain, irregular mesh) (a) Z (b) ∂Z∂X (c) ∂Z∂Y (d) M X (e) M Y (f) M XY . . . . . . 249 Difference between the approximate and the exact solutions (caseof the circular domain, irregular mesh) (a) Z (b) ∂Z∂X (c) ∂Z∂Y (d) M X (e) M Y (f) M XY . . . . . . . . . . . . . . . . 250 Values for approximate (bold line) and exact (thin line) solutions,their first order derivatives and the bending moments along seg-ment X = 1 (case of the square domain, irregular mesh) (a) Z (b) ∂Z∂X (c) ∂Z∂Y (d) M X (e) M Y (f) M XY . . . . . . . . . 351 Difference between approximate and exact solutions (case of thesquare domain, irregular mesh) (a) Z (b) ∂Z∂X (c) ∂Z∂Y (d) M X (e) M Y (f) M XY . . . . . . . . . . . . . . . . . . . . . . 352 An illustration for the construction of a D -dependency tree whena component of D -dependency graph has no dangling half-edge. . 753 An illustration for a sufficient condition for the regularity of bicu-bic in-plane parametrisation. . . . . . . . . . . . . . . . . . . . . 1454 An illustrations to proof of Theorem 6. . . . . . . . . . . . . . . 5455 An illustration for the ”Tangents Relation” . . . . . . . . . . . . . 6356 Different weight functions correspond to different in-plane parametri-sations of the boundary mesh elements. . . . . . . . . . . . . . . 63 Part I. Introduction
Definitions of the current Section are not always self-contained. The precisemathematical definitions of the notions used in this section will be given insubsequent sections,(example: the precise definition of G ( C )-continuity ,is inPart II). Input
Let
OXY Z a Cartesian coordinate system, ˜Ω a planar simply-connecteddomain lying in the XY -plane, together with its regular quadrangulation intoconvex elements ( by regular we mean that two elements have either an emptyintersection, an edge or a vertex in common .) The mesh may have an arbitrarytopological and geometrical structure (see Figure 1) and may have either apolygonal or a G -smooth piecewise-cubic B´ezier parametric global boundary. Output
Compute a D piecewise B´ezier tensor-product parametric surfaceof order n , so that (1) There is a one-to-one correspondence between the planar mesh elementsand the patches of the resulting surface.The orthogonal projection of every D patch onto the XY -plane defines a bijection between the patch and thecorresponding element of the planar mesh (see Figure 2). (2) The union of every two adjacent patches is G ( C )-regular. (3) The resulting surface is obtained as the constrained minimum for a givenenergy functional, relative to the cartesian plane ,( functional representa-tion of the surface.) (4)
One can impose additional interpolation/boundary constraints . For ex-ample, the resulting surface may be required to interpolate some D initialdata at vertices (see Figure 3). (5) Determine the dimension of the underlying approximation space.
An important result due to J. Peters [37] establishes that G continuityover any unstructured mesh implies actually C . Hence we are free to use G or C -continuity requirements , whichever is more adapted to the cases weconsider! We always suppose that the additional constraints noted above fit thesmoothness requirements . The surface we thus define or compute can be seenas a functional surface , that is a surface defined as Z = f ( X, Y ) over ˜Ω
The current work lies at the junction of Computer Aided Geometric Design andthe Finite Element Methods. Both fields have extremely extensive bibliographythat makes it impossible to present a full list of related works.
Problem definition , current sate and aim of the research Reviews and wide lists of references can be found, for example, in [13], [14],[21], [38], for CAGD methods) and in [8], [16], [46] (for Finite Element meth-ods). The recent Isogeometric Analysis (IGA) breakthrough [23] has broughttogether the fields of multi patch surface handling , higher order smooth orderapproximations and Finite Element Methods (FEM)..The related problems involve many ”parameters” like : type of the consid-ered meshes, choice of the functional space, the energy functional, smoothness, hence an abundant literature around the present subject.A classic problem in CAGD deals with the construction of piecewise para-metric B´ezier or B-spline surfaces or their extension to Non Uniform RationalB-Spline (NURBS) ,and it often requires (at least) G ( C )-continuity of theresulting surface. The researches may be subdivided into two large categories:the first category concentrates on the continuity conditions; the second categorystudies the different types of the energy (fairness) functionals. The purpose ofthe present review is to outline briefly some fundamental concepts of the relatedapproaches. We review mainly the case of quadrilateral patches and smooth sur-faces, and do not consider NURBS. One can onsider the construction of a smooth surfacesinterpolating a given 3 D mesh of curves (e.g. [30] [35], [39], [43], [44]) orstart with the construction of the mesh of curves which interpolates the givendata (e.g. [33], [41]). Usually, the initial triangular or quadrilateral mesh isnot required to be regular. However, it appears (see [35]) that the piecewiseparametric G -smooth interpolant does not necessarily exist for every mesh.Then either some restrictive assumption on the mesh of curves is introduced orsome modification of the mesh is made.Localizing the propagation of continuity constraints by refining surfaces isnecessary for some cases. Subdivision of some mesh elements (see [9], [10],[12], ) is commonly applied in order to improve the mesh quality (e.g. [35],[39], [42]) and to make the mesh admissible for interpolation by a smooth surface.Another techniques , [17], is based on macro-elements to keep a low order orderapproximation .Subdivision of an initial mesh element clearly implies that the resulting sur-face for the element is composed of several (polynomial/spline) pieces. The firststep is to check that the mesh of curves is admissible,next one proceeds withfilling the ”faces”. Both the weight functions and the inner control points in theunder-constraint situations are defined by application of some (usually local)heuristics, such as the least-square or averaging techniques (see, for example,[17], [33], [35], [42], and the references herein. Application of the local heuris-tics allows to construct a resulting surface by the local methods and to avoidany complicated computations.To avoid macro elements or subdivision one needs higher order tensor prod-uct patches for G construction of surfaces. The first candidate is the bi quarticpatch , but as we will show existence and uniqueness of a solution depends onthe underlying mesh structure . G construction of bi quintic B-spline surfacesover arbitrary topology is done in [50] and [49]. The aim is to simplify surface Problem definition , current sate and aim of the research representations by an approximation with such patches. To do that they derivemany local G properties similar to the one we will introduce. The authorsdo claim that the bi- quintic quadrilateral is of the lowest order possible, butdo not give any demonstration of this statement. They show that the prob-lem has no solution over general meshes for bi-cubic patches . Furthermore theactual dimension of the resulting basis is not studied and the functional usedfor approximating the given surfaces are not defined. Similarly, [20], constructa G surface by patch-by-patch scheme smoothly stitching bi-quintic B´ezierpatches . While the techniques described above are generally sufficient in orderto construct nicely looking surfaces,by approximation or interpolation of thegiven data, they usually require some preprocessing and the nature of the localheuristics may not reflect any geometrical characteristic of the resulting surface.[29] gives a higher order construction based on sixth order polynomials, we shallnot consider this here. A study of the energy functionals
The second category of techniques allowscontrolling the shape of the surface by minimisation of some (usually global)energy functional. The works, which study different forms of the energy func-tional, usually deal with intrinsically smooth functional bases (e.g. B-splinebasis) and in any case assume the regular structure of the mesh. The user is re-quired to enter only some essential interpolation data, the rest of the degrees offreedom are defined by the energy minimisation. The energies used in ComputerAided Geometrical Design commonly relate to the parametric representation ofthe surface and include the partial derivatives with respect to parameters. Thespectrum of the energies is very wide; the most advanced techniques computethe energies using some initial approximation of the resulting surface which leadto a good approximation of the ”natural” geometrical characteristics, such astotal curvature (e.g. ( [18], [45]).
CAGD and FEM are related domains, the main link being the two way ex-changes between geometry and meshes . Higher order approximations are oftenused in FEM, based on higher order polynomials ( p method ) triangular, regu-lar quadrilateral or so-called macro-elements ( splitting of convex quadrilaterals(see [9]) or triangles (see [10])), for details see [7], [16]. Moreover CADrepresentations have been used for the numerical solutions of partial differentialequations (PDEs), [25] , [40], and conversely some FEM methods have beenused for the design of geometrical objects , [31]. The real convergence is giventhe Isogeometric Analysis [23] , where PDEs are approximated by NURBS inthe physical space , using the geometric transformation that defines the domainand not the parametric( reference ) one. The Bivariate Triangular Spline Finite Elements
The construction of theBivariate Triangular Spline Finite Elements (BSFE) is closely connected to theapproach of the current work. The BSFE approach combines B´ezier-Bernsteinrepresentation of the polynomials and the requirement of C r ( r >
0) smoothness.It leads to the problem of determining minimal determining sets (MDS) . This
Problem definition , current sate and aim of the research will be at the center of the present work , so let us introduce this notion as itwas for BSFE.Let a triangulation of a simply-connected planar polygonal domain ˜Ω begiven. By definition, for integers n and 0 ≤ r ≤ n − space S rn consists of C r ( ˜Ω) smooth functions which are piecewise polynomials of total degree at most n over each triangle with respect to the barycentric coordinates. S rn is calleda bivariate spline finite element space. Note, that although for every trianglethe polynomials are represented in their B´ezier-Bernstein form with respect tothe barycentric coordinates, they also can be considered as polynomials in thefunctional sense.For a triangle with vertices A , B , C , the XY -coordinates of the B´eziercontrol points are given by ¯ P i,j,k = n ( iA + jB + kC ), i + j + k = n . Let Z ( ¯ P i,j,k ) denote the Z coordinate of the control point ˜ P i,j,k . Since at least C -continuity is assumed, the B´ezier control points of shared edges are unique,determining the dimension of the space S n .The dimension of S rn is given by the number of control points in a minimaldetermining set (MDS) , i.e. a minimal set of points nodal points D such that (see definition 5, below) : ∀ P i,j,k ∈ D, Z ( ¯ P i,j,k ) = 0 , and Z ∈ C r ⇒ Z ≡ S rn was initiated with a conjecture of Strangin [47]. The dimensionality depends on both the topological and geometricalstructure of the mesh; an arbitrary small perturbation of the mesh vertices maylead to changes in structure of the minimal determining set. The first importantresult was achieved by Morgan and Scott [32], with the dimension formula andexplicit basis for space S n , n ≥
5. The minimal determining sets (and thereforebases of the underlying spline spaces) were explicitly constructed for S for alltriangulations (see Alfred et all. [3]); S rn , n ≥ r + 2 for general triangulations(see [19], [22]); S rn , n = 3 r +1 for almost all triangulations (see [4]). We are notaware of results for the case r = 1 and n ≤ S rn is defined bytriangulation obtained by inserting the diagonals of each quadrilateral (see [27]).For the approximate solution of boundary-value problem, the spaces of type S r r , r ≥ X , Y , Z coor-dinates belongs to S rn - is given in [15]. There surfaces are build by interpolationand avoid the vertex enclosure problem (see below). However, parametrisationin the XY -plane can not be fixed a priory, which makes the approach unsuitablefor minimisation of the energy relating to the functional representation of thesurface. Problem definition , current sate and aim of the research The current work generalises the BSFE ap-proach on unstructured non degenerate convex quadrilateral meshing of a givenplanar domain ˜Ω with subparametric Bezi´er tensor-product ”Finite Elements”(FE) on each quadrilateral ( were we define an in plane parametrisation by abilinear mapping from a reference element.) This in-plane parametrisation leadsto the linearisation of the C -continuity conditions and reduces the problem toa linear constrained minimisation (see Part III). We also extend our results tothe case where the edge of a quadrilateral on the boundary of ˜Ω is given by acubic parametrisation.In addition, it provides a natural set up for explicitly computing the minimaldetermining set of the control points ˜ B ( n ) (see Subsection 1.3.2). We computethe ˜ B ( n ) for the space of C ( ˜Ω)-smooth, piecewise parametric polynomials ofdegree n ≥ C -smooth concatenationbetween adjacent patches are not local and the MDS depends on the topologyand geometry of the mesh. The resulting construction is done for all possibleunstructured mesh quadrangulations (both from a topological and geometricalpoints of view).The principal differences from the standard BSFE approach are the follow-ing. • The elements are defined over a square rather than a triangular referenceelement and has a tensor-product polynomial form. • Mapping between the reference element and the corresponding elementin ˜Ω is of at least of bilinear order. The resulting surface is given by afunctional minimisation or by interpolation. In classic BSFE the ”energies’are expressed in the parametric space, not in a ”physical” one. In ourcase, like for isoparametric finite elements , the space of functions doesnot coincide with the space of functional polynomials over quadrilaterals. • The MDS depends on the choice of mappings between the reference squareand the mesh elements. For a polygonal quadrilateral element the canonicbilinear mapping is used but a boundary mesh element with one curvilin-ear side requires a special analysis in order to choose the mapping in anoptimal way.The main contributions of the current work are listed below. • The current approach works for quadrilateral meshes with any valence forthe vertices (hence it can use standard FEM quad mesh generators .) • The MDS are constructed for both polygonal meshes and meshes which atthe boundary of ˜Ω consists of G -smooth piecewise B´ezier cubics . In the Problem definition , current sate and aim of the research last case, mappings between reference and boundary mesh elements areof higher order. Handling curved boundaries lead to better approximatesolutions of partial differential equations. • While the dimension of the MDS is uniquely defined, the choice of controlpoints, which participate in MDS ( i.e the basis functions), can be madein different ways. The current research analyses different MDS which aresuitable for different ”additional” conditions to cover the main types ofinterpolation and boundary conditions.The current work is restricted to an analysis of the MDS; it does not analysestability nor the approximation order of the solution, though the experimentalresults seem quite accurate.
A study of the continuity conditions
As noted above by [37] , on any quadri-lateral mesh imposing of G -continuity conditions is equivalent to the require-ment of C -smoothness. Hence, it is sufficient to analyse the G continuityconditions for the inner edges in terms of control points of adjacent patches.Moreover the study of G -continuity conditions for the control points adja-cent to a mesh vertex leads to results which fit the general Vertex Enclosuretheory (see Section 4). The results have elegant geometrical formulations,closely related to the structure of the planar mesh.In addition, a detailed analysis is made for the ”middle” control points adja-cent to an inner edge. All possible configurations of the adjacent mesh elementsare studied and the nice dependencies between the geometry of the elementsand the available degrees of freedom are defined. Choice of the energy functional
The shape of the resulting surface (in addi-tion to G -continuity and ”additional” constraints) is defined by minimisationof a ”natural physical” global energy functional, no local heuristics are used.Fixation of in-plane parametrisation a priory allows to define a functional formof energy, which makes the solution applicable to a wide range of problems inPDEs. Definition 1:
Consider the tensor product Bernstein-B´ezier polyomials of degree n over the unit square [ u, v ] ∈ [0 , . Let POL ( n ) , ˜ POL ( n ) and ¯ POL ( n ) be thecorresponding spaces ,obtained by defining, as coefficients of these polynomials,scalar, 2 D and 3 D control points respectively.We deal with B´ezier functions,(resp. plannar domains and surfaces) , followingthe CAGD conventions , [14], such objects will be defined as being of order n , (resp. ( n, m ) and ( n, m, r ) ), where n (resp. n, m , and n, m, r ) definesthe maximum (formal) degree(s) of a polynomial, the actual degree(s) may beless. For example by degree elevation of its first tensor term , a B´ezier bilinearquadrilateral [order (1 ,
1) ], can be considered as an order (2 , order or degree . Problem definition , current sate and aim of the research For a planar simply-connected domain ˜Ω, lying in XY -plane, and its quad-rangulation ˜ Q into non degenerate convex elements, the following definitionsand notations will be used. Definition 2:
Let m be some integer, a piecewise-polynomial 2 D function ˜Π isan order m global regular in-plane parametrisation of domain ˜Ω if (1) For every mesh element ˜ q ∈ ˜ Q , the restriction ˜ Q = ˜Π | ˜ q belongs to ˜ POL ( m ) and defines a regular mapping (see Definition 8) between the referencesquare and the planar element ˜ q . (2) For two adjacent mesh elements ˜ q and ˜ q (cid:48) , the 2 D control points of ˜ Q = ˜Π | ˜ q and ˜ Q (cid:48) = ˜Π | ˜ q (cid:48) coincide along the common edge of the elements.The space of all order m regular in-plane parametrisation will be denoted by˜ PAR ( m ) . Definition 3:
A piecewise-parametric 3 D function ¯Ψ agrees with a given globalin-plane parametrisation ˜Π if for every mesh element ˜ q ∈ ˜ Q the restriction ofthe function ¯ Q = ¯Ψ | ˜ q defines a mapping from the unit square into the 3 D space¯ Q : [0 , → R and the ( X, Y ) coordinates of ¯ Q coincide with the restrictionof the global in-plane parametrisation ˜Π | ˜ q . Definition 4:
Let n and m < n be some integers and ˜Π ∈ ˜ PAR ( m ) be somefixed degree m global regular in-plane parametrisation of domain ˜Ω. Thenspace ¯ FUN ( n ) ( ˜Π) is by definition composed of piecewise-parametric 3 D func-tions ¯Ψ so that (1) ¯Ψ agrees with the in-plane parametrisation ˜Π. (2) For every mesh element ˜ q ∈ ˜ Q , the Z -coordinate of the restriction ¯ Q = ¯Ψ | ˜ q belongs to POL ( n ) (and hence ¯Ψ | ˜ q is a subparametric FE). (3) ¯Ψ is a C -smooth function in the functional sense over ˜Ω: ¯Ψ ∈ C ( ˜Ω).It is important to note that space ¯ FUN ( n ) ( ˜Π) depends on the chosen in-planeparametrisation ˜Π, although in what follows it will be usually clear which un-derlying in-plane parametrisation is considered and the space will be usuallydenoted by ¯ FUN ( n ) . Definition 5:
Let ˜ CP ( n ) ( ˜Π) be a set of in-plane B´ezier control points of all patcheswhich result from degree elevation of a global regular in-plane parametrisation˜Π up to degree n for every patch. Since the B´ezier control points of the in-planeparametrisation always coincide along the shared edges, they are unambiguouslywell defined.A determining set D is a subset of ˜ CP ( n ) so that : ∀ P ∈ D, Z ( P ) = 0 ⇒ ¯Ψ ≡ Problem definition , current sate and aim of the research A determining set is called minimal determining set (MDS) ˜ B ( n ) if there is nodetermining which size is smaller.The subset ˜ B ( n ) depends on the chosen in-plane parametrisation ˜Π and is notnecessarily uniquely defined for a fixed in-plane parametrisation, but all in-stances have the same size, equal to the dimension of the vector space generatedby the corresponding basis functions.The purpose of the current work is to choose an in-plane parametrisationin some optimum way and describe the MDS ˜ B ( n ) for all n ≥ instances of ˜ B ( n ) .The principal goal of the work is to analyse the different instances of the MDSaccording to different ”additional” constraints.More details regarding different instances of the MDS, relations betweenMDS and the ”additional” constraints and other important definitions relevantfor the current approach are given in Section 6. Let a set of 3 D points be given and the topological structure of a 3 D quadri-lateral mesh be defined. The 3 D mesh itself is ”virtual” in the meaning thatthe quadrilateral faces of the mesh are defined in a symbolic manner, boundarycurves of 3 D patches are not specified.The goal is to construct such a G -smooth , piecewise B´ezier parametric sur-face that interpolates/approximates the given 3 D points and satisfies some op-tional additional conditions (for example normals of the tangent planes at themesh vertices or the boundary curve of the whole mesh may be specified ); theshape of the surface is defined (in addition to the interpolation/approximationconditions) by some energy functional which relates to the functional represen-tation of the surface Z = Z ( X, Y ).Define 3 D quadrilateral elements by connecting by straight segments theendpoints of every edge of the ”virtual” mesh (see Figure 3). If the constructionis such that the orthogonal projection of these 3 D elements on the XY -planedefines a bijection and forms a planar mesh of convex quadrilaterals then ouralgorithm can be applied.We consider in details the main kind of interpolation problems and providesa general approach, so that any interpolation/approximation problem can behandled in the same manner. The solution does not use the auxiliary construc-tion of a 3 D mesh of curves; the vertex enclosure constraints (see Section 4 )are intrinsically satisfied by the construction of the MDS, hence we can solvethe problem for any structure of the mesh.A comparison of the current approach and the standard techniques based oninterpolation o f 3 D mesh of curves is presented in Appendix, Section G. More detailed description of the possible kinds of additional constraints is given in Section5.2.1
Problem definition , current sate and aim of the research Consider 4th order partial differential equations (PDEs) (for example, the ThinPlate Problem or a biharmonc operator ): to have a conforming FEM one needsa C basis. An approximate solution can then be found by constrained minimi-sation of a corresponding energy functional (see [7] [16], [46]). Constraints(fixed explicitly or implicitly) are used for the imposition of boundary condi-tions , the energy functional is derived from the PDE and the computed solutioncan be represented as the functional representation of the resulting surface (anexample of the Thin Plate functional is given in Section 19).Usually subdivision of the domain into elements is done by classical 2Dmeshing techniques ( see for instance [6] ) and is included in the input ofthe problem together with the domain itself. The current research provides apossibility to construct a C -smooth piecewise-polynomial approximate solutionof a 4rth order partial differential equation for quadrilateral meshes with anarbitrary geometrical and topological structure , like the mesh shown in Figure 1.(Requirement of C -smoothness leads to a conforming approximate solution forfourth-order partial differential equations.) The solution is constructed and thedifferent boundary conditions are analysed for a planar mesh with a polygonalglobal boundary and for a planar mesh with piecewise-cubic G -smooth B´ezierparametric global boundary. Although the error estimation lies beyond thescope of the current work, practical results (see Section 19) show that theapproach leads to an approximate solution of a high quality. In the spirit ofIsogeometric Analysis one could also approximate 2nd order PDEs by mean ofthese C bases. Contents
Section 1 (Part I) describes the problem and the general approachto its solution, introduces essential concepts and formulates the principal goalsof the research. We compare the current research with related works, highlightsour contributions and describe the domains of application.Part II presents fundamental results and definitions from the common theoryof smooth surfaces, closely connected to the subject of the present work. Thegeneral method of solution, adopted in the current research, is described in PartIII. Two central Parts, IV and V , apply the general method for two differenttypes of planar mesh configurations. These parts contain the most importanttheoretical results, related to the existence and to the explicit construction ofthe solution, both in regular and in all possible degenerated cases. Full proofsof theoretical results as well as implementation algorithms are provided. PartVI shows that definitions based on the common fundamental concepts can benaturally generalised. The Generalisation leads to the composite solution witha wide range of applications.Part VII presents the computational examples, which allow to illustratethe correctness of the approach, and the results of application of the general
Problem definition , current sate and aim of the research solution to the Thin Plate problem. Part VIII discusses possible topics forfurther research.In order to free the main text from the technical details and long computa-tions as much as possible, all proofs, less important statements or statementswhich require a complicated algebraic formulation (Technical and AuxiliaryLemmas ) and some Algorithms are given in an Appendix (Sections D, Cand B respectively). In addition, the Appendix contains examples of the solu-tion of the partial differential equation (Section A) and several Sections thatincludes some auxiliary material (Sections E, F and G). Figures.
Generally, the Figures are placed in the end of every Section. Fig-ures which present results of the practical application of the approach are givenin Appendix, Section A. The full list of figures is given at the beginning of thework.
Notations.
A special Section (Section 2) presents a list of the most commonand useful formal notations and definitions. In addition, Section 2 contains anindex list for some essential definitions used in the current work.
Fonts and underlines.
Tems and notions ,with a precise definition, are usually written in italic fontand/or appear in quotes. (For example, the ”Middle” system of equations orthe middle control points).Fig. 1: Irregular quadrilateral mesh for a circular domain.
Problem definition , current sate and aim of the research Fig. 2: 3 D B´ezier parametric patch and the corresponding planar mesh element.Fig. 3: Planar mesh and 3 D surface interpolating 3 D points and tangent planesat the mesh vertices. Notations Here is presents a list of the most common and useful formal notations anddefinitions. ¯ A = ( ˜ A, A ) = ( A X , A Y , A ) - a 3 D point or vector, where ˜ A = ( A X , A Y ) is its2 D component in XY -plane and A is its Z -coordinate. In general, ¯ willbe used for 3 D objects and ˜ for 2 D objects in XY -plane. || ¯ A || or || ˜ A || - Euclidean norm of 3 D or 2 D vector. < ¯ A, ¯ B > - cross product of two 3 D vectors. < ˜ A, ˜ B > - Z -coordinate of the cross product of two vectors lying in XY -plane(signed length of the cross product of two 2 D vectors).( ¯ A, ¯ B ) - scalar product of two 3 D vectors. mix ¯ A ¯ B ¯ C = ( ¯ A, < ¯ B, ¯ C > ) = det A X , A Y AB X B Y BC X C Y C - ”mixed” product of three3 D vectors. B ni ( u ) = (cid:16) ni (cid:17) u i (1 − u ) n − i , u ∈ [0 , i = 0 , . . . , n - degree n Bernstein polyno-mial of one variable. B nij ( u, v ) = (cid:16) ni (cid:17) (cid:16) nj (cid:17) u i (1 − u ) n − i v j (1 − v ) n − j , ( u, v ) ∈ [0 , , i, j = 0 , . . . , n -degree n tensor-product Bernstein polynomial of two variables. P ( u ) = (cid:80) ni =0 P i B ni ( u ) - B´ezier polynomial of degree n , P i , ˜ P i , ¯ P i - B´ezier control points ( in 1 D, D, D ). deg ( P ) - actual degree of a polynomial; the lowest integer such that P ( u ) canbe represented in the form P ( u ) = (cid:80) deg ( p ) i =0 α i u i , with α deg ( p ) (cid:54) = 0 . P ( u, v ) = (cid:80) ni,j =0 P ij B nij ( u, v ) - tensor-product B´ezier polynomial of order n , P ij (or ˜ P i,j , ¯ P i,j ) - B´ezier control points (see Figure 2). Notations ˜ A, ˜ B, ˜ C, ˜ D - four vertices of a convex quadrilateral planar mesh element (seeFigure 4).˜ t ( ˜ A, ˜ B, ˜ C, ˜ D ) = ˜ A − ˜ B + ˜ C − ˜ D - twist characteristic of the element, doubledifference between the midpoints of the diagonals of the quadrilateral (seeFigure 4). Vertices, edges and twist characteristics of two adjacent mesh elements ˜ λ, ˜ λ (cid:48) , ˜ γ, ˜ γ (cid:48) , ˜ ρ, ˜ ρ (cid:48) - vertices of two adjacent planar mesh elements (see Figure 5).˜ e ( R ) = ˜ ρ − ˜ γ , ˜ e ( C ) = ˜ γ (cid:48) − ˜ γ , ˜ e ( L ) = ˜ λ − ˜ γ - directed planar mesh edgesadjacent to vertex ˜ γ (see Figure 5).˜ t ( R ) = ˜ t (˜ γ, ˜ ρ, ˜ ρ (cid:48) , ˜ γ (cid:48) ), ˜ t ( L ) = ˜ t (˜ γ, ˜ γ (cid:48) , ˜ λ (cid:48) , ˜ λ ), where ˜ t is the twist characteristic ofthe planar mesh elements (see Figure 6). Edges and twist characteristics of elements adjacent to a given vertex
Letplanar mesh elements share the common vertex ˜ V of degree val ( V ) (see Figure7)˜ e ( j ) = ˜ V ( j ) − ˜ V ( j = 1 , . . . , val ( V )) - directed planar mesh edges adjacent tovertex ˜ V , ordered counter clockwise.˜ t ( j ) = ˜ V − ˜ V ( j ) + ˜ F ( j ) − ˜ V ( j +1) ( j = 1 , . . . , val ( V ) for an inner vertex and j = 1 , . . . , val ( V ) − V . Let two adjacent planar mesh elements be parametrized by ˜ L ( u, v ) and ˜ R ( u, v )respectively and let ˜ L (1 , v ) ≡ ˜ R (0 , v ) (see Figure 8).˜ L v = ˜ R v = ∂ ˜ L∂v (1 , v ) = ∂ ˜ R∂v (0 , v ) - partial derivatives of the in-plane parametri-sations in the direction along the common edge.˜ L u = ∂ ˜ L∂u (1 , v ) , ˜ R u = ∂ ˜ R∂u (0 , v ) - partial derivatives of the in-plane parametri-sations along the common edge in the cross direction.˜ λ i , ˜ ρ i - coefficients of the polynomials ˜ L u and ˜ R u with respect to the B´ezierbasis.˜ λ ( power ) i , ˜ ρ ( power ) i - coefficients of the polynomials ˜ L u and ˜ R u with respect tothe power basis. c ( v ) , l ( v ) , r ( v ) - scalar weight functions from the definition of G -continuity(see Definition 7). Note that these are B´ezier functions. c j , l j , r j - coefficients of the weight functions with respect to the B´ezier basis. Notations c ( power ) j , l ( power ) j , r ( power ) j - coefficients of the weight functions with respect tothe power basis.( order ( c ) , order ( l ) , order ( r )) (or ( deg ( c ) , deg ( l ) , deg ( r ))) - match which is de-fined by formal (or actual) degrees of the weight functions. max deg ( l, r ) - maximum among deg ( r ) and deg ( l ). D data of the resulting surface ¯ C j , ¯ L j , ¯ R j ( j = 0 , ..., n ) - for two adjacent patches, B´ezier control points alongthe common edge and of the rows adjacent to the common edge in the leftand the right patch respectively. Control points ¯ C j are called ”central”control points and control points ¯ L j , ¯ R j are called ”side” control points(see Figure 9).∆ ¯ C j , ∆ ¯ L j , ∆ ¯ R j - first-order differences of the control points (see Figure 9)∆ ¯ C j = ¯ C j +1 − ¯ C j , j = 0 , . . . , n −
1∆ ¯ L j = ¯ C j − ¯ L j , j = 0 , . . . , n ∆ ¯ R j = ¯ R j − ¯ C j , j = 0 , . . . , n (1) Let ˜ V be a planar mesh vertex of valence val ( V ) and let ˜ V have at least oneadjacent inner edge. The following notations are used for the control pointsadjacent to the vertex and participating in G -continuity conditions for at leastone inner edge (see Figure 10)¯ V - control point corresponding to the mesh vertex, the control point (or itscomponents) will be called V -type control point.¯ E (1) , . . . , ¯ E ( val ( V )) for an inner vertex or ¯ E (2) , . . . , ¯ E ( val ( V ) − for a boundaryvertex - the first control points adjacent to ¯ V along the inner edges ema-nating from the vertex; these control points (or their components) will becalled tangent or E -type control points.¯ D (1) , . . . , ¯ D ( val ( V )) for an inner vertex ¯ D (2) , . . . , ¯ D ( val ( V ) − for a boundaryvertex - the second control points adjacent to ¯ V along the inner edges em-anating from the mesh vertex; these control points (or their components)will be called D -type control points.¯ T (1) , . . . , ¯ T ( val ( V )) for an inner vertex or ¯ T (1) , . . . , ¯ T ( val ( V ) − for a boundaryvertex, which are adjacent to ¯ V and do not lie at any edge; these controlpoints (or their components) will be called twist or T -type control points. Let twoadjacent patches be parametrized as shown in Figure 11 and let the parametri-sations agree along the common edge.
Notations ¯ (cid:15) ( R ) , ¯ (cid:15) ( C ) , ¯ (cid:15) ( L ) - the first-order derivatives along right, central and left edgescomputed at vertex ¯ V .¯ (cid:15) ( R ) = ∂ ¯ R∂u (0 ,
0) = n ∆ ¯ R ¯ (cid:15) ( C ) = ∂ ¯ R∂v (0 ,
0) = ∂ ¯ L∂v (1 ,
0) = n ∆ ¯ C ¯ (cid:15) ( L ) = − ∂ ¯ L∂u (1 ,
0) = − n ∆ ¯ L (2)¯ τ ( R ) - the second-order mixed partial derivatives of the left and the rightpatches computed at vertex ¯ V .¯ τ ( R ) = ∂ ¯ R∂u∂v (0 ,
0) = n (∆ ¯ R − ∆ ¯ R )¯ τ ( L ) = − ∂ ¯ L∂u∂v (1 ,
0) = − n (∆ ¯ L − ∆ ¯ L ) (3) δ ( C ) = ∂ R∂v (0 ,
0) = ∂ L∂v (1 ,
0) = n ( n − C − ∆ C ) = n ( n − C − C + C ) - Z -component of the second-order partial derivative along the central edgecomputed at vertex ¯ V . Partial derivatives of all patches sharing a common vertex
Let patchessharing a common vertex ¯ V be parametrized as shown in Figure 10 and letparametrisations of adjacent patches agree along the common edges.¯ (cid:15) ( j ) = n ( ¯ E ( j ) − ¯ V ) = ∂ ¯ P ( j − ∂v (0 ,
0) = ∂ ¯ P ( j ) ∂u (0 ,
0) - first-order partial derivativein the direction of edge ˜ e ( j ) , computed at vertex ¯ V .¯ τ ( j ) = n ( ¯ V − ¯ E ( j ) + ¯ T ( j ) − ¯ E ( j +1) ) = ∂ ¯ P ( j ) ∂u∂v (0 ,
0) - second-order mixed partialderivative of patch ¯ P ( j ) computed at vertex ¯ V . δ ( j ) = n ( n − D ( j ) − E ( j ) + V ) = ∂ P ( j − ∂v (0 ,
0) = ∂ P ( j ) ∂u (0 , Z -component of the second-order partial derivative in the direction of edge˜ e ( j ) , computed at vertex ¯ V . Definitions of some sets of control points ˜ B ( n ) , ˜ B ( n ) G , ˜ B (4 , - see Definitions 5, 10 and 24 respectively.˜ CP ( n ) , ˜ CP ( n ) G , ˜ CP ( n ) F , ˜ CP (4 , - see Definitions 5, 10, 10 and 23 respectively. middle control points - see Definition 16 for a mesh with a polygonal globalboundary and Lemma 22 and Definition 21 for a mesh with a smoothglobal boundary. Definitions of some functional spaces ¯ FUN ( n ) , ¯ FUN (4 , - see Definition 4 and Part VI respectively.˜ PAR ( m ) - see Definition 2. S rn - see Subsection 1.2.2. Notations Definitions of some special equations and systems of equations ”Eq(s)” indexed equation - see general Definition 9, Lemma 5 for a mesh witha polygonal global boundary and Lemma 19 for a mesh with a smoothglobal boundary. ”Eq(s)-type” equations - see Definition 9. ”Middle” system of equations - see Definition 16 for a mesh with a polygonalglobal boundary and Lemma 22 and Definition 21 for a mesh with asmooth global boundary. ”Restricted Middle” system of equations - see Definition 21. ”sumC-equation”,”C-equation” - see Definition 20 .
Notations Fig. 4: A planar mesh element.Fig. 5: Two adjacent planar mesh elements.
Notations Fig. 6: Vectors important for computation of the actual degrees of the weight functionsin the case of the bilinear parametrisation of two adjacent mesh elements.Fig. 7: Planar mesh elements adjacent to the common vertex.
Notations Fig. 8: Partial derivatives of in-plane parametrisations for two adjacent patches.Fig. 9: B´ezier control points adjacent to the common edge of two patches.
Notations Fig. 10: B´ezier control points adjacent to some mesh vertex.Fig. 11: Two adjacent parametric patches with G -smooth concatenation alongthe common boundary. Part II. Some fundamental resultsregarding G -smooth surfaces Construction of the MDS (defined in Subsection 1.3.2) is based on the analysisof the smoothness conditions between adjacent patches. The current Sectioncontains the formal definitions of the different kinds of smoothness and presentsthe general theoretical results of the vertex enclosure problem, which are closelyconnected to the analysis of the local structure of the MDS.
Two kinds of smoothness - functional and parametric ones - will be involved.The following standard definitions are used.
Definition 6 ( C -smoothness of a functional surface): A functional surface Z ( X, Y )is C -smooth over domain ˜Ω if for every point ( X, Y ) ∈ ˜Ω the first-order partialderivatives ∂Z∂X ( X, Y ), ∂Z∂Y ( X, Y ) are well defined and continuous over ˜Ω.In order to define G parametric smoothness, let us consider two adjacentquadrilateral patches ¯ L ( u, v ) and ¯ R ( u, v ). Let every one of the patches beparametrized by the unit square (see Figure 11), such that their parametrisa-tions agree along the common edge and the concatenation between the patchesis G -smooth (continuous)¯ L (1 , v ) = ¯ R (0 , v ) for every v ∈ [0 ,
1] (4)In addition, every patch is supposed to be sufficiently smooth, with at least acontinuous first order partial derivatives along the common edge . Equation 4implies that¯ L v (1 , v ) = ¯ R v (0 , v ) for every v ∈ [0 ,
1] (5)
Definition 7 ( G -smooth concatenation between two parametric patches): Patches¯ L ( u, v ) and ¯ R ( u, v ) join G -smoothly along the common edge if and only if thereexist a scalar-valued weight functions l ( v ), c ( v ), r ( v ) such that for every v ∈ [0 , L u (1 , v ) l ( v ) + ¯ R u (0 , v ) r ( v ) + ¯ L v (1 , v ) c ( v ) = 0 (6) l ( v ) r ( v ) < < ¯ L u (1 , v ) , ¯ L v (1 , v ) > (cid:54) = 0 (8)(see Definition given in [35]).Geometrically < ¯ L u , ¯ L v > and < ¯ R u , ¯ R v > define the tangent plane normalsfor the left and the right patches respectively. Equation 6 means that thetangent planes of the adjacent patches are co-planar along the common edge.Equation 8 means normal to the the tangent plane does not vanish and Equation7 controls the orientation of the patches in order to avoid cusps. We have thefollowing Lemma : The vertex enclosure problem Lemma 1:
Let two patches with a linear common edged join G smoothly thentheir respective parametrisation join C continuously. Proof
The common face of the two patches being a linear segment, one cantrivially reparametrize one as image of the second ,this is a special case Peters’fundamental Lemma [37]. (cid:116)(cid:117)
Lemma 1
It is then possible to combine the functional representation of the surfacedefining the energy functional and the parametric representation of the surfacein order to impose the G smoothness constraints in parametric form. The current Section mainly relates to the work [35], devoted to smooth in-terpolation of a given 3 D mesh of curves. The work is chosen as the mainreference, because it formulates and analyses in details the general vertex enclo-sure constraint. The satisfaction of the vertex enclosure constraints determinesthe existence of a G -smooth interpolant for a given 3 D mesh of curves. (Gen-eralisation of the vertex enclosure problem to the case of concatenation of a fewpatches around a common vertex with a definite degree of smoothness can befound in [36].) Let a 3 D mesh of polynomial curves be given, ( we only study meshes whichfaces are 4-sided,) construct a G -smooth piecewise B´ezier tensor-product in-terpolanting these curves. ( [35] contains a full analysis for the mixed tri-angular/quadrilateral meshes and shows that from a theoretical point of viewthe quadrilateral or triangular form of a patch does not lead to essential differ-ences for the vertex enclosure constraint). In the problem formulated above, theboundary curves of every patch (mesh curves) are given and the inner B´eziercontrol points of every patch play the role of unknowns. These unknowns shouldsatisfy G -continuity constraints, which means that the weight functions fromDefinition 7 should exist for every inner edge of the mesh. In particular thesefunctions should exist for every one of the edges that share a common innervertex. Consideration of G -continuity constraints together for all edges ad-jacent to the same vertex leads to a so called vertex enclosure problem. Thevertex enclosure constraint is met at a mesh vertex ¯ V if weight functions couldbe simultaneously defined for each mesh edge emanating from ¯ V .For an inner mesh vertex Equations 6 applied to all edges emanating fromthe vertex have a circulant nature (the ”left” patch of the ”first” edge is alsothe ”right” patch of the ”last” edge) and lead to a linear system of equationssuch that the matrix of the system has a circulant structure. Independently ofthe order and geometry of the mesh curves, the matrix is always invertible atthe odd vertices and rank deficient at the even vertices. At the even verticesthe rank of the matrix is equal to its size minus one, which generally meansthat one additional constraint for every even mesh vertex should be satisfied inorder to allow a G -smooth interpolation. A mesh is called admissible if a G The vertex enclosure problem smooth interpolant can be constructed (or in other words, if weight functionsfor all inner edges can be defined without contradictions).In Peters, [35], the vertex enclosure constraint is considered in its mostgeneral form (for example, the mesh curves sharing the common vertex may havedifferent polynomial degrees), which leads to quite complicated equations. Theconstraint is not written explicitly, sufficient conditions that allow concludingthat a given mesh is admissible are supplied.We will show that in our case, the explicit form of the vertex enclosureconstraint becomes very simple and elegant. The Subsection presents formulasfrom Peters [35] in order to verify later that results of the current work fit thegeneral theory. The general results are formulated in notations of the currentwork (see Section 2). Although it makes the presentation quite different fromits original, the conversion between different forms of presentation is purelytechnical and straightforward. In order to make the formulas more compactand clear, some minor simplifying assumptions, which are always satisfied inthe current work, will be used.According to the problem definition, a quadrilateral mesh of curves of de-gree m should be G -smoothly interpolated by piecewise tensor-product B´ezierpatches of degree n . G -continuity between a pair of adjacent patches impliesthat the following two equations should be satisfied. ”Tangent Constraint” c ∆ ¯ C + l ∆ ¯ L + r ∆ ¯ R = 0 (9) ”Twist Constraint” ( n − c ∆ ¯ C + deg ( c ) c ∆ ¯ C + n l ∆ ¯ L + deg ( l ) l ∆ ¯ L + n r ∆ ¯ R + deg ( r ) r ∆ ¯ R = 0 (10)Here notations from Section 2 are used. In particular ¯ C j , ¯ L j , ¯ R j are the controlpoints of B´ezier patches (see Figures 11) and c j , l j , r j are coefficients of theweight functions.In the interpolation problem formulated in work [35], tangents ∆ ¯ C , ∆ ¯ L ,∆ ¯ R and boundary curve control points ¯ C j ( j = 0 , . . . , n ) are given, and twistcontrol points ¯ L and ¯ R as well as coefficients of the weight functions serve asunknowns.For a vertex ¯ V with val ( V ) emanating curves, superscript j will be used whentangent or twist constraint is considered for the curve with order number j =1 , . . . , val ( V ). Control points ¯ C ( j ) i , ¯ L ( j ) i , ¯ R ( j ) i for i = 0 , V , ¯ E ( j ) and ¯ T ( j ) will be used respectively for the vertex, tangent andtwist control points (see Subsection 2.6.2).The ”Tangent Constraint” defines (up to a scale factor) the zero-indexedcoefficients of the weight functions. These coefficients depend on the geometryof the tangent vectors of curves emanating from the vertex.The ”Twist Constraint” for curve with order number j can be rewritten inthe form m l ( j )0 ¯ T ( j ) − m r ( j )0 ¯ T ( j − = ¯ A ( j ) (11) The vertex enclosure problem where ¯ A ( j ) = ( n − c ( j )0 ∆ ¯ C ( j )1 + deg ( c ( j ) ) c ( j )1 ∆ ¯ C ( j )0 + l ( j )0 (cid:16) m ¯ E ( j ) − ( m − n )∆ ¯ L ( j )0 (cid:17) + deg ( l ( j ) ) l ( j )1 ∆ ¯ L ( j )0 − r ( j )0 (cid:16) m ¯ E ( j ) + ( m − n )∆ ¯ R ( j )0 (cid:17) + deg ( r ( j ) ) r ( j )1 ∆ ¯ R ( j )0 (12)For an inner vertex ¯ V , the ”Twist Constraint” applied simultaneously to all the val ( V ) edges emanating from the vertex, leads to a circulant linear system ofequations. M T = A (13)Here matrix M has a circulant structure M = l (1)0 . . . − r (1)0 − r (2)0 l (2)0 . . . − r (3)0 l (3)0 . . . . . . l ( val ( V ) − . . . − r ( val ( V ) − l ( val ( V ) −
00 0 0 . . . − r ( val ( V ))0 l ( val ( V ))0 (14)and T = ¯ T (1) ...¯ T ( val ( V )) A = ¯ A (1) ...¯ A ( val ( V )) (15)Equation 13 together with the dependency defined by the ”Tangent Constraint’ lead to the general Parity Phenomenon. The following two theorems are due toPeters [35]. Theorem 1 (The General Parity Phenomenon):
For an inner vertex ¯ V , matrix M (see Equation 13) is of full rank if and only if ¯ V is an odd vertex. Otherwiseits rank is equal to val ( V ) − G -smooth interpolant depends on the solvabilityof a quite complicated equation in terms of coefficients of the weight functions.It appears that in order to conclude that a given mesh of curves is admissible ,the complicated equation should not be solved explicitly. Instead of it, one canverify whether the mesh satisfies the sufficient conditions formulated below. The vertex enclosure problem Theorem 2 (Sufficient conditions for the vertex enclosure constraint):
If at every in-ner even vertex ¯ V of a given mesh of curves either of the following holds • (Colinearity Condition) The vertex is a 4-vertex and all odd-numberedand all even-numbered curves emanating from V have colinear tangentvectors (that is the tangent vectors form an ’X’). • (Sufficiency of C Data) The mesh curves emanating from ¯ V are compat-ible with a second fundamental form at ¯ V .then the mesh is admissible. Part III. General linearisation method
This Part shows that the functional space ¯
FUN ( n ) (see Definition 4) reduces theproblem formulated in paragraph 1.1 to the solution of some linear constrainedminimisation problem. Later the general method for linearisation will be appliedto cases when a planar mesh has a polygonal (Part IV) or piecewise-cubic G -smooth global boundary (Part V).In addition this Part provides some important definitions and notations re-lated to the general flow of the solution and to the analysis of the MDS for thedifferent kinds of ”additional” constraints. Properties of a global regular in-plane parametrisation (see Definition 2) playa principal role in the following discussion. We now complete this definitionnby :
Definition 8: (Regular parametrisation of a mesh element)
In-plane parametrisation ˜ P ( u, v ) = ( P X ( u, v ) , P Y ( u, v )), ( u, v ) ∈ [0 , of asingle mesh element ˜ p ∈ ˜ Q is called regular if and only if (1) ˜ P ( u, v ) is a bijective mapping between the unit square and ˜ p (2) ˜ P ( u, v ) is at least C -smooth (3) The Jacobian J ( ˜ P ) ( u, v ) of the mapping has no singular points: for every( u, v ) ∈ [0 , det ( J ( ˜ P ) ( u, v )) = det (cid:18) ∂P X ∂u ∂P Y ∂u∂P X ∂v ∂P Y ∂v (cid:19) (cid:54) = 0 (16) Theorem 3:
Let ˜Π be a fixed global regular in-plane parametrisation and ¯Ψa piecewise parametric 3 D function agreeing with ˜Π (see Definition 3). Lettwo adjacent mesh elements be parametrized as shown in Figure 11, and let¯ L = ( ˜ L, L ) and ¯ R = ( ˜ R, R ) denote the restriction of ¯Ψ on the elements. Thenthe G -continuity of ˜Π along the common edge is equivalent to the satisfactionof the equation L u ( v ) l ( v ) + R u ( v ) r ( v ) + L v ( v ) c ( v ) = 0 (17)where c ( v ) = < ˜ L u , ˜ R u >, l ( v ) = < ˜ R u , ˜ L v >, r ( v ) = − < ˜ L u , ˜ L v > (18)are fixed scalar-valued functions and L u ( v ) , R u ( v ) , L v ( v ) are the Z -componentsof the partial derivatives along the common edge.Functions c ( v ), l ( v ), r ( v ) computed according to Equation 18 will be calledconventional weight functions. Linearisation of the minimisation problem Proof
See Appendix, Section D.Lemma 1 and Theorem 3 imply that the requirement of C -smoothness indefinition of space ¯ FUN ( n ) (Definition 4, Item (3) ) may be substituted by theweaker requirement of G -smoothness, and, furthermore, by the requirementthat Equation 17 is satisfied for every inner edge. Thus the smoothness condi-tions can be studied in terms of the Z -components of the B´ezier control pointsof the adjacent patches. Moreover, we have the following Lemma . Lemma 2:
Let ˜Π ∈ ˜ PAR ( m ) and ¯Ψ ∈ ¯ FUN ( n ) ( ˜Π) for n > m . Then the sum inEquation 17 is a B´ezier polynomial of (formal) degree n +2 m −
1; its coefficientsare linear functions in terms of Z -components of the control points. In order tosatisfy the G -continuity condition, it is sufficient to impose N umEqF ormal = n + 2 m linear equations of the form : (cid:88) j + k = s ≤ j ≤ n ≤ k ≤ m − (cid:16) nj (cid:17) (cid:16) (cid:17) ( l k ∆ L j + r k ∆ R j ) + (cid:88) j + k = s ≤ j ≤ n − ≤ k ≤ m (cid:16) n-1j (cid:17) (cid:16) (cid:17) c k ∆ C j = 0 (19)where s = 0 , . . . , n + 2 m −
1. Here ∆ L j , ∆ R j ( j = 0 , . . . , n ) and ∆ C j ( j = 0 , . . . , n −
1) are first order differences between Z -components of the controlpoints, defined in Subsection 2.6.1.The system may contain redundant equations. The number of necessary andsufficient equations is given by N umEqActual = n + max { max deg ( l, r ) + 1 , deg ( c ) } (20) Proof
See Appendix, Section D.
Definition 9:
The following notations will be used • Indexed equation ”Eq(s)” - the equation which follows from equality tozero of the B´ezier coefficient with index s • NumInd - the number of the indexed equations (for one edge). Althoughin most situations
N umInd is equal to the total number of equations,the equality should not be necessarily satisfied, because one may like toseparate some equations with a special meaning from the homogeneoussystem of the indexed equations. • ”Eq(s)” -type equations - pair of equations ”Eq(s)” and ”Eq(NumInd-1-s)” ( s = 0 , . . . , (cid:100) ( N umInd − / (cid:101) ).Definition of ”Eq(s)” -type equations clearly makes sense in cases when equations ”Eq(s)” and ”Eq(NumInd-s)” are symmetric (for example, in the case when in-plane parametrisations of the elements adjacent to an edge have a symmetricform and both vertices of the edge are inner). The formal definition of ”Eq(s)” -type equations in non-symmetric cases will also be used.Of course, the linear systems should be considered for all inner edges. Theglobal system may have a sufficiently complicated structure: some control points Linearisation of the minimisation problem participate in G -continuity equations for more than one edge. In addition, arelatively high degree of in-plane parametrisation results in high degrees of theweight functions and increases both the total number of equations and thecomplexity of each equation.A study of the global system is equivalent to a study of the MDS, since it iscomposed of the control points, which correspond to free variables of the linearsystem. In addition to an analysis of the dimensionality, one should clearly becareful of a ”uniform distribution” of the basic control points. Dimensionalityand structure of the MDS for concrete choices of in-plane parametrisation willbe analysed in detail in Parts IV and IV. Special attention will be paid to astudy of the geometrical meaning of equations involved in the linear system. The current Subsection describes the commonly used types of interpolation andboundary constraints and shows which control points become fixed as a resultof application of the constraint. The relation between MDS and the chosen typeof ”additional” constraints will be explained in greater detail in Subsection 6.2.
The following constraints (the first one and optionally the second or/and thethird ones) are usually applied in the case of an interpolation problem. (Vertex)-interpolation.
The resulting surface should pass through the given3 D point at every mesh vertex. The constraint involves V -type control points(see Subsection 2.6.2 for definition) of the mesh vertices. (Tangent plane)-interpolation. The normal of the tangent plane at every 3 D vertex should have a specified direction. At every mesh vertex, in addition to V -type control point, the constraint involves E -type control points. The constraintautomatically fits the requirement of G -smoothness. The assignment values fortwo E -type control points of two non-colinear edges at every vertex is sufficient. (Boundary curve)-interpolation. The resulting surface should interpolate agiven 3 D curve along the global boundary. The constraints result in assignmentvalues for all control points lying on the global boundary of the mesh.The following notations will be used in order to specify the kind of interpola-tion problem: round brackets () mean that the type of interpolation is applied,square brackets [] mean that the type of interpolation is optional. For exam-ple, the (vertex)[tangent plane]-interpolation problem means that the resultingsurface should pass through given 3 D points at vertices and in addition thenormals of the tangent planes at vertices might be specified. The following standard boundary conditions are imposed when an approximatesolution of some partial differential equation should be found. Here ¯ P = ( ˜ P , P )is the restriction of the resulting function to some boundary mesh element ˜ p ∈ ˜ Q Linearisation of the minimisation problem Simply-supported boundary condition.
The standard simply-supported bound-ary constraint implies that P ( boundary ) = 0 (21)should be explicitly fixed. Let ¯ P ( u, v ) be a patch, such that ˜ P ( u,
0) lies onthe global boundary of domain Ω (see Figure 12),then the simply-supportedboundary condition means that P i = 0 for every i = 0 , . . . n (22) Clamped boundary condition.
The standard clamped boundary condition meansthat P ( boundary ) = 0 and ∂P∂ ˜ N ( boundary ) = 0 (23)where ˜ N is the unit planar normal to the boundary of the domain. Let patch¯ P ( u, v ) have a regular in-plane parametrisation ˜ P ( u, v ). Then ∂P∂ ˜ N = ∂P∂v (cid:16) ∂v∂P X N X + ∂v∂P Y N Y (cid:17) = ∂P∂v || ˜ P u || det ( J ˜ P ) (24)where || ˜ P u ( boundary ) || (cid:54) = 0, because otherwise ˜ N ( boundary ) is not correctlydefined. Condition ∂P∂v ( boundary ) = 0 together with condition P ( boundary ) = 0imply equality to zero of the Z -components of two rows of the boundary controlpoints (see Figure 12) P ij = 0 for every i = 0 , . . . n, j = 0 , F and G , consider P ( boundary ) = F and ∂P∂ ˜ N ( boundary ) = G .If functions F and G are represented or approximated by B´ezier parametricpolynomials of some degrees (for example degree of F should be less or equal tothe chosen degree of polynomial for Z -component of the patch), then this generalcondition does not lead to additional complications. For the current approachit is important that the control points which are defining the boundary condi-tions be free from dependencies which follow from G -smoothness conditions. Asimply-supported boundary condition affects the control points along the globalboundary of domain; a clamped boundary condition affects the control pointsalong the global boundary of the domain and the control points adjacent to theboundary. Let ˜Π be a fixed global regular parametrisation, ¯Ψ ∈ ¯ FUN ( n ) ( ˜Π) and ¯ P =( ˜ P , P ) = ¯Ψ | ˜ p be the restriction of ¯Ψ on some mesh element ˜ p ∈ ˜ Q . The in-planeparametrisation ˜ P = ˜Π | ˜ p is fixed, therefore all partial derivatives of Z withrespect to X and Y become linear in terms of Z -components of B´ezier controlpoints. Any energy functional defined as the integral of some quadratic expres-sion of the partial derivatives has a quadratic form in terms of Z -componentsof the control points, hence we have a quadratic minimisation problem. Linearisation of the minimisation problem The expression for the energy functional depends on the chosen in-planeparametrisation of the mesh element and may be different for different elements.Although the basic kind of parametrisation considered in the present work (thebilinear parametrisation) leads to very simple formulas, a separate computationis generally required for every mesh element.Section E (see Appendix) presents an example of computation of energyfunctional in the case of bilinear in-plane parametrisation.Fig. 12: Control points involved into the boundary conditions.
Principles of construction of MDS To clarify the discussion, and restrict their number that should be analysed weintroduce some subsets of control points.
Definition 10:
Let (see Figures 13 and 14)˜ CP ( n ) F : subset of in-plane control points ˜ CP ( n ) which are not involved in the G -continuity conditions;˜ CP ( n ) G ( ˜ B ( n ) G ) : subset of in-plane control points ˜ CP ( n ) ( resp . minimal deter-mining set ˜ B ( n ) ) which participate in G -continuity conditions.The set ˜ CP ( n ) F of control points (in other words, all control points which do notlie at some inner edge or adjacent to it) clearly belong to any determining set. Lemma 3:
Dimensions of the subsets ˜ CP ( n ) G and ˜ CP ( n ) F are given by the followingformulas (here | | denotes dimension of a set) | ˜ CP ( n ) F | = ( n − | F ace inner | + ( n − n − | F ace boundarynon − corner | +( n − | F ace corner || ˜ CP ( n ) G | = | V ert non − corner | + 3 | V ert boundarynon − corner | + (3 n − | Edge inner | (26)The following relations take place | ˜ CP ( n ) | = | ˜ CP ( n ) G | + | ˜ CP ( n ) F | , | ˜ B ( n ) | = | ˜ B ( n ) G | + | ˜ CP ( n ) F | (27)The important conclusion from Lemma 3 is that we only need to study thestructure and dimensionality of ˜ B ( n ) G - subset of the minimal determining setwhich participates in G -continuity conditions . The definition of paragraph 1.1, implies that any ”additional” constraints isassumed to be consistent and to fit the G -continuity requirements.An ”additional” constraint result in some definite control points being fixed.These control points get their values according to the ”additional” constraintsand can not influence the satisfaction of G continuity conditions. Definition 11:
The minimal determining set ˜ B ( n ) is said to fit a given ”addi-tional” constraint if any control point which should be fixed according to this”additional” constraint either Principles of construction of MDS • Belongs to ˜ B ( n ) or • Does not belong to ˜ B ( n ) but depends only on the control points whichbelong to ˜ B ( n ) An MDS is called ”pure” if it is constructed according to G -conditions aloneand is not required to fit any specific ”additional” constraint. As it was mentioned above, MDS is not uniquely defined. According to thecurrent approach, construction of the MDS will be built up gradually and willfollow two (closely connected) kinds of locality concepts.At every step of MDS construction, some subset of the linear equation willbe considered. The first principle of locality requires that the subset includes in-dexed equations (see Definition 9) with successive indices. In addition, the anal-ysis starts from the application of the equations locally, for example, to edgessharing some common vertex or to control points participating in G -continuityequation for a given edge. Control points, which get their status (basic or de-pendent) during the construction step, clearly obey the principle of geometricallocality. The local set of the control points which are classified according to thelocal application of some set of equations, is called ”local template” of MDS (seeFigure 23).As soon as the local analysis is completed, one should define the order inwhich the local templates should be constructed and take care to put togetherthe local templates without contradiction (different local templates may inter-sect!). Construction of the MDS implies assignment of the definite status to every oneof the control points. Control points, which belong to the MDS, are basic con-trol points and the rest are dependent control points. For a given ”additional”constraint, the basic control points which get their values according to the ”ad-ditional” constraints are called basic fixed , the remaining basic control pointsare basic free .Classification process by definition includes- Construction of the minimal determining set MDS (or several instances ofthe MDS).- Description of the dependency of every one of the dependent control pointson the basic ones. (More precisely, dependency of Z -component corre-sponding to the dependent control point on Z -components correspondingto the basic control points).- For a given ”additional” constraint, the choice of the instance of MDSthat fits the constraint. From MDS to solution of the linear minimisation problem It is important to note, that although usually several different configurationsof MDS are considered, construction of MDS follows some definite principles(see Subsection 6.3) and of course does not cover all possible configurations.According to the current approach, in case none of the constructed instances of˜ B ( n ) fits some ”additional” constraint, MDS of a higher degree will be consid-ered. However, failure to choose a suitable instance of MDS does not necessarilyimply that a ”pure” algebraic solution of the constrained linear system does notexist in space ¯ FUN ( n ) . For example, it will always be assumed that any V -type control point belongs to MDS, while an algebraic solution may use such acontrol point as a dependent one in non-interpolating problems.In order to make the discussion precise, the following definition of the stages of the classification process is introduced. Definition 12:
A Stage is usually a large part of the classification process, whichis defined by some set of equations and so that at the end of the stage: (1)
All control points which participate (or may participate under definitegeometrical conditions) in these equations are classified (as basic or de-pendent ones) and the status of every one of the control points is final, itcan not be changed during the next stages of the classification. (2)
Any dependent control point depends only on the control points with thefinal basic classification status. (3)
All these equations are satisfied by classification of the control points.
As soon as for a given ”additional” constraint, a suitable MDS is constructed,dependencies of the dependent control points are defined and energy for everymesh element is computed, construction of the solution of the linear minimi-sation problem is made in straightforward algebraic manner (see Appendix,Section F for more details).
From MDS to solution of the linear minimisation problem (a) (b) (c) Fig. 13: Control points, which do not participate in G -continuity conditions(a) Inner element (b) Corner element (c) Boundary non-corner element.Fig. 14: Control points of a global in-plane parametrisation. Part IV. MDS for a quadrilateral mesh witha polygonal global boundary
The following minor mesh limitations are always supposed to be satisfied- The mesh consists of strictly convex quadrilaterals. Every mesh elementis a convex quadrilateral and angle between any two sequential edges isstrictly less than π .- Boundary vertices have valence 2 (a corner vertex) or 3 (see Figures 15(a),15(b)). The situation shown in Figure 15(c) is not allowed.- Any inner edge has at most one boundary vertex.The limitations are naturally satisfied in most of the practical situations. Inaddition, a standard technique of necklacing (see [24]) may be applied to amesh in order to achieve the second and the third requirement.It is convenient to introduce an additional minor mesh limitation, which isrequired to be satisfied only when MDS of degree n = 4 is considered. In thiscase a planar mesh should satisfy the ”Uniform Edge Distribution Condition”,defined as follows Definition 13:
The mesh is said to satisfy the ”Uniform Edge Distribution Con-dition” if for any even vertex of degree ≥ (a) (b) (c) Fig. 15: An illustration for the mesh limitations.
In-plane parametrisation Fig. 16: Examples of meshes, which do not satisfy the ”Uniform Edge Distribu-tion Condition” . For a quadrilateral planar mesh element, the bilinear in-plane parametrisationwill be considered. It is a natural choice because for a general quadrilateral(without curvilinear sides) it clearly provides a parametrisation of the min-imal possible degree, which finally leads to the minimal possible number of G -continuity equations.For a convex quadrilateral planar mesh element with vertices ˜ A, ˜ B, ˜ C, ˜ D (seeFigure 4) the parametrisation is given by explicit formula˜ P ( u, v ) = ˜ A (1 − u )(1 − v ) + ˜ Bu (1 − v ) + ˜ Cuv + ˜ D (1 − u ) v (28)and det ( J ( ˜ P ) ( u, v ) > u, v ) ∈ [0 , , hence the following Lemmaholds. Lemma 4:
For a strictly convex planar quadrilateral element, the bilinear in-plane parametrisation of the element is regular.The bilinear parametrisations for all mesh elements clearly satisfies therequirements of Definition 2 and define a degree 1 global regular in-planeparametrisation ˜Π ( bilinear ) ∈ ˜ PAR (1) .Explicit formulas for in-plane control points which belong to ˜ CP ( n ) G ( ˜Π ( bilinear ) )are given in Technical Lemma 1 (see Appendix, Section C).
10 Conventional weight functions and linear form of G -continuity conditions Application of the general linearisation method (Theorem 3 and Lemma 2) toa particular case of bilinear in-plane parametrisation leads to the next Lemma.
Lemma 5:
Let two adjacent mesh elements with vertices ˜ λ, ˜ λ (cid:48) , ˜ γ, ˜ γ (cid:48) , ˜ ρ, ˜ ρ (cid:48) (seeFigure 5) each with a bilinear in-plane parametrisation, then (1) The conventional weight functions l ( v ) , r ( v ) and c ( v ) along the commonedge are B´ezier polynomials of (formal) degrees 1,1 and 2 respectively and their coefficients with respect to the B´ezier basis depend on the geometryof the planar elements in the following way l = (cid:104) ˜ ρ − ˜ γ, ˜ γ (cid:48) − ˜ γ (cid:105) c = (cid:104) ˜ ρ − ˜ γ, ˜ λ − ˜ γ (cid:105) l = (cid:104) ˜ γ − ˜ γ (cid:48) , ˜ ρ (cid:48) − ˜ γ (cid:48) (cid:105) c = (cid:104) (cid:104) ˜ ρ − ˜ γ, ˜ λ (cid:48) − ˜ γ (cid:48) (cid:105) − (cid:104) ˜ λ − ˜ γ, ˜ ρ (cid:48) − ˜ γ (cid:48) (cid:105) (cid:105) r = −(cid:104) ˜ γ (cid:48) − ˜ γ, ˜ λ − ˜ γ (cid:105) c = −(cid:104) ˜ λ (cid:48) − ˜ γ (cid:48) , ˜ ρ (cid:48) − ˜ γ (cid:48) (cid:105) r = −(cid:104) ˜ λ (cid:48) − ˜ γ (cid:48) , ˜ γ − ˜ γ (cid:48) (cid:105) (29) (2) The system of n +2 linear equations ”Eq(s)” for s = 0 , . . . , n +1 is sufficientin order to satisfy the G - continuity condition along the common edge” Eq ( s )” ( n + 1 − s )( l ∆ L s + r ∆ R s ) + s ( l ∆ L s − + r ∆ R s − )+ ( n − s )( n +1 − s ) n c ∆ C s + s ( n +1 − s ) n c ∆ C s − + s ( s − n c ∆ C s − = 0 (30)Here ∆ L j , ∆ R j for j < j > n and ∆ C j for j < j > n − l ( v ), r ( v ) and c ( v ) may have lower actual degrees than 1, 1and 2. Weight function l ( v ) becomes constant if ˜ t ( R ) is parallel to ˜ γ (cid:48) − ˜ γ ; r ( v )becomes a constant if ˜ t ( L ) is parallel to ˜ γ (cid:48) − ˜ γ and the actual degree of c ( v ) isat most 1 if ˜ t ( R ) and ˜ t ( L ) are parallel (see Figure 6 and Subsection 2.3.1 fordefinition of ˜ t ( R ) and ˜ t ( L ) ). For example, in the case of two adjacent squareelements deg ( l ) = deg ( r ) = 0, deg ( c ) ≤
1. It implies that n + 1 linear equationsare sufficient in order to guarantee G -smooth concatenation and therefore anadditional degree of freedom is available.
11 Local MDS
As stated in Subsection 6.3, construction of the MDS follows the principleof locality. All control points are subdivided into several types: V , E , D and T -type control points adjacent to some mesh vertex (see Subsection 2.6.2) and the middle control points adjacent to some mesh edge (see Definition 16). Everytype of the control points is responsible for the satisfaction of some definitesubset of the linear equations.The current Subsection is devoted to an analysis of equations applied toa separate mesh vertex or edge, a possible influence of the other equations isignored. The analysis results in construction of local MDS, templates, whichlocally define which control points belong to MDS and describe the dependenciesof the dependent control points. The same set of equations may define severalstructures of the MDS, suitable for the different mesh geometry and differenttypes of ”additional” constraints. Definition 14:
We say that two local templates are different, if they contain adifferent number of basic control points or if there is a difference in the typesor a principal difference in the location of the control points.
Note 1:
Sometimes, the templates do not uniquely specify which control pointsshould be classified as basic. In case of ambiguity, classification of the control points is made arbitrarily. The local geometric characteristics, such as edgelengths or angles, plays an important role in stabilizing the solution and can bea matter of additional research. E , V -type control points for aseparate vertex based on ” Eq (0)” -type equations Formal substitution of s = 0 in Equation 19 gives” Eq (0)” l ∆ L + r ∆ R + c ∆ C = 0 (31)It is precisely the general ”Tangent Constraint” (Equation 9) applied to Z -components of the control points. The difference is that in the curve meshinterpolation problem 3 D tangents ∆ ¯ L , ∆ ¯ R , ∆ ¯ C are given and coefficientsof the weight functions are unknown. In the current case on the contrary,coefficients of the weight functions are fixed a priory and Z -components of thecontrol points play the role of unknowns. ”Eq(0)” -type equations have a very simple geometrical meaning. Let ˜ V be aplanar mesh vertex of degree val ( V ) and ˜ e ( j ) ( j = 1 , . . . , val ( V )) be a directedplanar mesh edges emanating from ˜ V (see Figure 7). Then for the edge ˜ e ( j ) ,zero-indexed coefficients of the weight functions can be rewritten as follows l ( j )0 = (cid:104) ˜ e ( j − , ˜ e ( j ) (cid:105) , r ( j )0 = −(cid:104) ˜ e ( j ) , ˜ e ( j +1) (cid:105) , c ( j )0 = (cid:104) ˜ e ( j − , ˜ e ( j +1) (cid:105) (32)Note that ¯ E ( j − − ¯ V , ¯ E ( j ) − ¯ V , and ¯ E ( j +1) − ¯ V are colinear if and only if0 = mix ¯ E ( j − − ¯ V ¯ E ( j ) − ¯ V ¯ E ( j +1) − ¯ V = n mix ˜ e j − E ( j − − V ˜ e j E ( j ) − V ˜ e j +1 E ( j +1) − V = l ( j )0 ( E ( j +1) − V ) + c ( j )0 ( E ( j ) − V ) + r ( j )0 ( E ( j − − V ) (33)which exactly means that the ”Eq(0)” -type equation for the edge with ordernumber j is satisfied, thus ¯ E ( j ) − ¯ V , ( j = 1 , . . . , val ( V ) ) are coplanar ; thiscan be summed up in the lemma: Lemma 6:
Let ¯ V be a 3 D vertex control point and let ¯ E ( j ) ,( j = 1 , . . . , val ( V ))be the edge control points adjacent to the vertex (see Subsection 2.6.2). Thenfor ”Eq(0)” -type equations applied simultaneously to all edges sharing vertex¯ V , the tangent vectors ¯ E ( j ) − ¯ V ( j = 1 , . . . , val ( V )) should be coplanar. At every vertex, the tangent plane is defined by any three noncolinear controlpoints lying in it. Let V -type control point and such a pair of E -type controlpoints ˜ E ( i ) , ˜ E ( j ) (1 ≤ i, j ≤ val ( V )), that ˜ e ( i ) and ˜ e ( j ) are not colinear, be clas-sified as basic . Any other E -type control point ˜ E ( k ) is classified as dependent. Its dependency (dependency of the corresponding Z -component) is defined bysystem of ”Eq(0)” -type equations and has the following explicit form E ( k ) = (cid:104) ˜ e ( i ) , ˜ e ( j ) (cid:105) (cid:8) − E ( i ) (cid:104) ˜ e ( j ) , ˜ e ( k ) (cid:105) − E ( j ) (cid:104) ˜ e ( k ) , ˜ e ( i ) (cid:105) + V (cid:0) (cid:104) ˜ e ( i ) , ˜ e ( j ) (cid:105) + (cid:104) ˜ e ( j ) , ˜ e ( k ) (cid:105) + (cid:104) ˜ e ( k ) , ˜ e ( i ) (cid:105) (cid:1)(cid:9) (34) For any inner vertex, we classify as basic any V type control point that has theproperties above.The remaining E -type control points depend on the basic control pointsaccording to Equation 34. The correspondent local template is shown in Figure17(a).This local MDS clearly fits all types of considered ”additional” constraints,including the (vertex)(tangent plane)-interpolation condition. The basic controlpoints can be easily classified into free and fixed, depending on the kind of the”additional” constraints. According to the mesh limitations, any non-corner boundary vertex ˜ V has ex-actly one adjacent inner edge ˜ e (2) .The following two local templates are defined: T B ( V,E ) (Figure 17(b)). The local MDS contains ˜ V , the boundary controlpoint ˜ E (1) and the inner control point ˜ E (2) . This template is always usedwhen the boundary edges are colinear. T B ( V,E ) (Figure 17(c)). The local MDS contains ˜ V and two boundary controlpoints ˜ E (1) , ˜ E (3) . This template is always used in the case of boundarycurve interpolation and simply supported boundary conditions, providedthe boundary edges are not colinear.The following two examples show that for the considered ”additional” constraintat least one of T B ( V,E ) , T B ( V,E ) provides the local MDS which fits the con-straint and present classification of the basic control points into free and fixed. (Vertex)(Boundary curve)-interpolation condition . If the boundary edgesare not colinear, then T B ( V,E ) is used. ˜ V , ˜ E (1) and ˜ E (3) are basic fixed controlpoints; ˜ E (2) is dependent, the corresponding Z -component is computed accord-ing to Equation 34. If the boundary edges are colinear, then T B ( V,E ) is used.˜ V and ˜ E (1) are basic fixed control points , ˜ E (2) is a basic free one. In thiscase, one should verify that the data of the boundary curve fits the ”TangentConstraint” : the given value of E (3) should be equal to the value computedaccording to Equation 34, using the given values of V and E (1) . Clamped boundary condition . It is always possible to make use of
T B ( V,E ) .All basic control points are fixed. The standard clamped boundary constraintclearly satisfies the ”Tangent Constraint” . In case of a more complicated clampedboundary condition, classification of the control points remains unchanged. One should verify that the boundary condition and the ”Tangent Constraint” fit to-gether. Value of E (3) computed according to Equation 34, should be equal tothe value given by the boundary condition. D , T -type control points for aseparate vertex based on ” Eq (1)” -type equations In the current Subsection it will always be assumed that V , E -type control pointsare classified and ”Eq(0)” -type equations are satisfied by choice of an appropri-ate template. Substitution of s = 1 in Equation 30 leads to the formula” Eq (1)” n ( l ∆ L + r ∆ R ) + ( l ∆ L + r ∆ R )+( n − c ∆ C + 2 c ∆ C = 0 (35)This is a particular case of the general ”Twist Constraint” (Equation 10) appliedto Z -components of the control points. An advantage of the current particularcase is that the coefficients of the weight functions have a clear geometricalmeaning, closely connected to the structure of the initial planar mesh. It allowsrewriting ”Eq(1)” in a more meaningful form.Let ˜ V be a planar mesh vertex of degree val ( V ) and ˜ e ( j ) ( j = 1 , . . . , val ( V ))be directed planar mesh edges emanating from ˜ V (see Figure 7).Let ¯Ψ ∈ ¯ FUN ( n ) ( ˜Π ( bilinear ) ) and patch ¯ P ( j ) denote the restriction of ¯Ψ onthe mesh element adjacent to ˜ V , containing edges ˜ e ( j − and ˜ e ( j ) (see Figure10) as part of its boundary. The following important relations between XY -components of the first and second-order partial derivatives of the patches andthe initial mesh data hold¯ (cid:15) ( j ) = (˜ e ( j ) , n ( E ( j ) − V ))¯ τ ( j ) = (˜ t ( j ) , n ( D ( j ) − E ( j ) + V )) = (˜ t ( j ) , nn − δ ( j ) ) (36)Here ¯ (cid:15) ( j ) and ¯ τ ( j ) , δ ( j ) are the first and second order partial derivatives (seeSubsection 2.6.3) and ˜ e ( j ) , ˜ t ( j ) are directed planar edges and twist character-istics of the planar mesh elements (see Subsection 2.3.1). Relations given inEquation 36 allow to conclude that the following Lemma holds. Lemma 7:
Let all ”Eq(0)” -type equations for all inner edges adjacent to vertex˜ V be satisfied by classification of V and E -type control points. Then for inneredge ˜ e ( j ) , ”Eq(1)” -type equation applied to the control points adjacent to vertex˜ V , has the following geometrical form tw ( j ) + tw ( j − = coef f ( j ) δ ( j ) (37)Where tw ( j ) = 1 (cid:104) ˜ e ( j ) , ˜ e ( j +1) (cid:105) mix ¯ τ ( j ) ¯ (cid:15) ( j ) ¯ (cid:15) ( j +1) , coef f ( j ) = (cid:104) ˜ e ( j − , ˜ e ( j +1) (cid:105)(cid:104) ˜ e ( j − , ˜ e ( j ) (cid:105)(cid:104) ˜ e ( j ) , ˜ e ( j +1) (cid:105) (38) Proof
See Appendix, Section D.It is important to note, that tw ( j ) and δ ( j ) are linear expressions in termsof V -type, E -type and T -type or D -type control points; tw ( j ) contains a single non-classified T -type point T ( j ) and δ ( j ) contains a single non-classified D -typepoint D ( j ) . Application of Lemma 7 to all edges emanatingfrom a common inner vertex leads to the following Theorem.
Theorem 4:
Let ˜ V be an inner vertex of degree val ( V ) and let ”Eq(0)” -typeequations for all edges adjacent to the vertex be satisfied. Then (1) The system of ”Eq(1)” -type equations applied simultaneously to all edgesadjacent to ˜ V has the following form M tw (1) ... tw ( val ( V )) = coef f (1) δ (1) ... coef f ( val ( V )) δ ( val ( V )) (39)where M is the matrix with a simple circulant structure M = . . . . . . . . . . . . (40) (2) In the case of an even odd vertex matrix M is of full rank. In the case ofan odd even vertex rank ( M ) = val ( V ) − Circular Constraint ” val ( V ) (cid:88) j =1 ( − j coef f ( j ) δ ( j ) = 0 (41) Note 2:
The ”Circular Constraint” does not involve T -type control points. Itestablishes some dependency between D -type control points adjacent to a givenvertex. (Under the assumption that all V -type and E -type control points arealready classified according to the first stage of the classification process).Results of Theorem 4 clearly fit the general Parity Phenomenon (Theorem1). The ”Circular Constraint” corresponds to the necessary condition whichshould be satisfied for the right sides of the general ”Twist Constraint” (seeEquation 12) for an even vertex. The main advantage of the current partic-ular case is a very elegant and geometrically meaningful form of the ”CircularConstraint” . The way in which the ”Circular Constraint” is applied presents the secondimportant difference between the current approach and the standard techniquesfor interpolation by 3 D smooth piecewise parametric surface. Usually some ini-tial data (3 D mesh of curves in work [35]) is tested to satisfy the necessarycondition. In case of negative answer, a G -smooth surface cannot be con-structed. In the current approach one may take advantage of the fact that evenin the case of (vertex)(tangent plane)-interpolation, a boundary curve of twoadjacent patches (which has at least degree 4) is not totally fixed. At least onecontrol point in the middle of every curve remains non-fixed. The purpose is toconstruct the MDS in such a manner, that every vertex at which the ”CircularConstraint” should be satisfied, has at least one ”own” basic D -type controlpoint. Note 3:
Coefficient coef f ( j ) of δ ( j ) ( D ( j ) ) in the ”Circular Constraint” may beequal to zero; it happens if the planar mesh edges ˜ e ( j − and ˜ e ( j +1) are colinear.In this case D ( j ) does not contribute to the ”Circular Constraint” . Definition 15 (Regular -vertex): Vertex of valence 4 is called 4-regular if 4 planaredges emanating from the vertex form two colinear pairs: ˜ e (1) is colinear to ˜ e (3) and ˜ e (2) is colinear to ˜ e (4) . Lemma 8:
Regular 4-vertex is the only possible configuration of the edges adja-cent to some inner even vertex when all coefficients coef f ( j ) ( j = 1 , . . . , val ( V ))are equal to zero and the ”Circular Constraint” is satisfied automatically. Proof of the Lemma
The strict convexity of the mesh elements implies thatany inner even vertex ˜ V has degree val ( V ) = 4 at least.Let coef f ( j ) = 0 for every j = 1 , . . . , val ( V ). In particular, coef f (2) = 0and so ˜ e (1) and ˜ e (3) are colinear and lie on some straight line ˜ l (1 , ; coef f (3) = 0and so ˜ e (2) and ˜ e (4) are colinear and lie on some straight line ˜ l (2 , (see Figure18). Therefore for val ( V ) = 4 the vertex is proven to be regular.It remains to show that val ( V ) could not be greater than 4. Indeed, let val ( V ) >
4. Then, ˜ e (2) should be colinear to both ˜ e (4) and ˜ e ( val ( V )) (becauseboth coef f (3) and coef f (1) are equal to zero). But ˜ e (4) and ˜ e ( val ( V )) can not becolinear because ˜ e ( val ( V )) lies strictly between ˜ e (4) and ˜ e ( deg (1)) which span anangle less than π due to the strict convexity of the mesh elements. (cid:116)(cid:117) Lemma 8 . Some necessary and sufficient conditions for the satisfaction of the ”CircularConstraint” at a separate inner even vertex
Results of the current paragraphcorrespond to the sufficient vertex enclosure conditions formulated in Theorem2. Although the results do not contribute directly to the construction of theMDS, they provide an additional confirmation that the present approach fitsthe general theory of G -smooth piecewise parametric surfaces.Lemma 8 from the previous paragraph shows that for an inner vertex ofdegree 4 colinearity of two pairs of emanating edges is a sufficient condition forthe satisfaction of the ”Circular Constraint” . Necessary and sufficient conditionsare presented in the following Lemma. Lemma 9:
Let ˜ V be an inner even vertex (1) If δ ( j ) , j = 1 , . . . , val ( V ) ( Z -components of the second-order derivativesin the directions of the planar edges) are chosen in such a manner thatthey are compatible at ˜ V with second-order partial derivatives of somefunctional surface, then the ”Circular Constraint” is satisfied. (2) For a non-regular 4-vertex ˜ V , compatibility of δ ( j ) , j = 1 , . . . , ”CircularConstraint” . Proof
See Appendix, Section D.
Note 4:
Lemma 9 does not mean that the resulting surface is necessarily C -smooth at the vertex. Besides values of δ ( j ) ( j = 1 , . . . , val ( V )) there is always atleast one additional degree of freedom ( T -type control point) which implies thatthe second-order partial derivatives in the functional sense are not necessarilywell defined at the vertex. A local template for an inner odd vertex is shown in Figure 19(a).All D -type control points are classified as basic and all T -type control pointsare dependent. There are val ( V ) basic control points in all.The correctness of the classification and dependencies of T -type controlpoints are explained below.As stated in Theorem 4, matrix M is invertible for an odd inner vertex.Therefore all D -type control points can be classified as basic and T -type con-trol points depend on them (and on V -type and E -type basic control points)according to equation tw (1) ... tw ( val ( V )) = M − coef f (1) δ (1) ... coef f ( val ( V )) δ ( val ( V )) (42)In greater detail, D -type control points together with V -type and E -type basiccontrol points fully define values of δ ( j ) ( j = 1 , . . . , val ( V )). Equation 42 definesdependency of tw ( j ) ( j = 1 , . . . , val ( V )) on δ ( j ) ( j = 1 , . . . , val ( V )), and finallythe values of T -type control points are given by T ( j ) = n (cid:104) ˜ e ( j ) , ˜ e ( j +1) (cid:105) tw ( j ) − V (cid:16) n (cid:104) ˜ t ( j ) , ˜ e ( j +1) − ˜ e ( j ) (cid:105) < ˜ e ( j ) , ˜ e ( j +1) (cid:105) (cid:17) + E ( j ) (cid:16) n (cid:104) ˜ t ( j ) , ˜ e ( j +1) (cid:105) < ˜ e ( j ) , ˜ e ( j +1) (cid:105) (cid:17) + E ( j +1) (cid:16) − n (cid:104) ˜ t ( j ) , ˜ e ( j ) (cid:105) < ˜ e ( j ) , ˜ e ( j +1) (cid:105) (cid:17) (43)The classification may be formally subdivided into two steps : at the first stepall D -type control points are classified as basic, at the second step all T -typecontrol points are classified as dependent and their dependencies are established. Even vertex excluding the regular -vertices The local template for an innerodd vertex, excluding regular 4-vertices, is shown in Figure 19(b). val ( V ) − D -type control points and one T -type control point are classified as basic; val ( V ) basic control points in all. D -type control of some edge may be chosento be dependent only if two neighboring edges of the edge are not colinear.The correctness of the classification and dependencies of the dependent con-trol points are explained below.According to Theorem 4, the ”Circular Constraint” should be imposed on D -type control points adjacent to an inner even vertex, excluding regular 4-vertices. Classification of D -type and T -type control points can be made asfollows.At the first step D -type control points are classified. One D -type controlpoint with a non-zero coefficient, say ˜ D ( k ) , is chosen. The remaining D -typecontrol points are classified as basic and D ( k ) depends on them (and V , E -typebasic control points) according to the ”Circular Constraint” D ( k ) = 2 E ( k ) − V + n ( n − coeff ( k ) (cid:80) val ( V ) j = 1 j (cid:54) = k ( − j + k +1 coef f ( j ) δ ( j ) (44)At the second step T -type control points are classified. Rank-deficiency ofmatrix M means that one of T -type control points, for example ˜ T ( val ( V )) , can beclassified as a basic control point. The remaining T -type control points dependon this control point and D -type control points (which are classified during theprevious step) according to the following equation tw (1) tw (2) ... tw ( val ( V ) − = (cid:18) M , . . . , val ( V ) − , . . . , val ( V ) − (cid:19) − coef f (1) δ (1) − tw ( val ( V )) coef f (2) δ (2) ... coef f ( val ( V ) − δ ( val ( V ) − (45)where M , . . . , val ( V ) − , . . . , val ( V ) − is a ( val ( V ) − × ( val ( V ) −
1) square matrix whichcontains val ( V ) − M . Regular -vertex Local template for a regular 4-vertex is shown in Figure19(c). All D -type control points and one T -type control point are classified asbasic, the remaining T -type control points are dependent; there are val ( V ) + 1basic control points in all.In case of a regular 4-vertex the ”Circular Constraint” should not be explic-itly imposed. At the first step all D -type control points are classified as basic.At the second step one of the T -type control points is classified as basic and oth-ers as dependent ones. T -type control points do not depend on D -type controlpoints; dependency between T -type control points is defined by the relation tw (1) = − tw (2) = tw (3) = − tw (4) (46) In case of a boundary non-corner vertex there is exactly one adjacent inner edge˜ e (2) and so it is sufficient to consider a single constraint in the form of Equation37. The following two local templates are defined: T B ( D,T ) (Figure 20(a)). The local MDS contains a D -type control point˜ D (2) and one T -type control points ( ˜ T (1) ). This template can always beused , excluding the case when the boundary edges are not co-linear andthe clamped boundary condition is imposed. T B ( D,T ) (Figure 20(b)). The local MDS contains two T -type control points˜ T (1) and ˜ T (2) . This template is used only in the case when the boundaryedges are not co-linear and the clamped boundary condition is imposed.Below the different ”additional” conditions are considered and T B ( D,T ) isshown to fit almost all situations. Any kind of Interpolation/Simply-supported boundary condition.
These boundary conditions do not involve neither D -type nor T -type controlpoints adjacent to the boundary vertex. While D -type control point do notcontribute to Equation 37 in case of colinear boundary edges, the coefficient of T (1) is equal to − n (cid:104) ˜ e (2) , ˜ e (3) (cid:105) and never vanishes. Therefore it is very natural tochoose a T -type control point as a dependent variable (template T B ( D,T ) ), itsdependency is defined by Equation 37. Both basic control points are classifiedas free control points. Clamped boundary condition.
The clamped boundary condition involves both T -type control points anddo not involves D -type control point. Therefore a MDS which fits the clampedboundary condition can not define T -type control point as dependent on D -typecontrol point (see 11).If the boundary edges are colinear, D -type control point do not participatein Equation 37. Equation 37 defines dependency between T (1) and T (2) tw (1) + tw (2) = 0 (47)In case of the standard clamped boundary condition (Equation 23) tw (1) = tw (2) = 0 and the dependency is automatically satisfied. Any other clampedboundary condition should be checked to be compatible with the G -smoothnessrequirement. Template T B ( D,T ) can be used: the basic D -type control pointis classified as free and the basic T -type control point is classified as fixed.If the boundary edges are not colinear, one should use template T B ( D,T ) (template T B ( D,T ) does not longer fit the ”additional” condition). Both T -typebasic control points are fixed. Dependency of D (2) -type is defined according toEquation 37 D (2) = 2 E (2) − V + n ( n − coeff (2) ( tw (1) + tw (2) ) (48) middle control points for aseparate edge Definition 16:
Let the global bilinear in-plane parametrisation be considered.For an inner edge, the set of n − ”Eq(s)” for s =2 , . . . , s = n − ”Middle” system of equations.Control points ˜ L , . . . , ˜ L n − , ˜ R , . . . , ˜ R n − and ˜ C , . . . , ˜ C n − are respectivelycalled the ”side” middle and the ”central” middle control points. Let all V , E , T and D -type control points be already classified and all ”Eq(0)” -type and ”Eq(1)” -type equations be satisfied. The only non-classified controlpoints which participate in G -continuity equations for an inner edge are the middle control points and the ”Middle” system is not yet satisfied. We shallshow that the middle control points are sufficient in order to satisfy the ”Middle” system of equations for an inner edge. It provides the possibility to define thelocal MDS separately for every inner edge. The following geometric conditionplays a principal role in the analysis of the ”Middle” system of equations. Definition 17:
For two adjacent patches with vertices ˜ λ ,˜ λ (cid:48) ,˜ γ ,˜ γ (cid:48) ,˜ ρ ,˜ ρ (cid:48) (see Fig-ure 21), let (˜ λ − ˜ γ ) ( proj ) , (˜ λ (cid:48) − ˜ γ (cid:48) ) ( proj ) , (˜ ρ − ˜ γ ) ( proj ) , (˜ ρ (cid:48) − ˜ γ (cid:48) ) ( proj ) denotethe lengths of projections of the corresponding planar vectors onto directionperpendicular to ˜ γ (cid:48) − ˜ γ . (In other words, distances from vertices ˜ λ ,˜ λ (cid:48) ,˜ ρ ,˜ ρ (cid:48) to theline (˜ γ, ˜ γ (cid:48) )) • It will be said that the ”Projections Relation” holds if (˜ ρ − ˜ γ ) ( proj ) (˜ ρ (cid:48) − ˜ γ (cid:48) ) ( proj ) = (˜ λ − ˜ γ ) ( proj ) (˜ λ (cid:48) − ˜ γ (cid:48) ) ( proj ) (49) • For the bilinear global in-plane parametrisation ˜Π ( bilinear ) , the ”Projec-tions Relation” means that the coefficients of the conventional weight func-tions l ( v ), r ( v ) satisfy the following equation: l r − r l = 0 (50)The structure of the local MDS depends on the geometrical configuration oftwo adjacent mesh elements. The following two local templates for classificationof the middle control points are defined. T M n − T M n − L j , ˜ R j ), j = 2 , . . . , n −
2. The template is applied if the ”ProjectionsRelation” holds.Theorem 5 justifies the choice of the local MDS and proves the correctness ofthe classification of the control points.
Theorem 5:
Let ˜ L and ˜ R be the restrictions of the global in-plane parametrisa-tion ˜Π ( bilinear ) ) on two adjacent mesh elements. Let ”Eq(0)” -type and ”Eq(1)” -type equations be satisfied for the common edge and all E , V , D , T -type controlpoints be classified. Then for any n ≥ (1) Consistency. The ”Middle” system of equations has a solution in termsof the middle control points. (2) Classification of the middle control points. The following classifica-tion of the middle control points guarantees that the ”Middle” system ofequations for an inner edge is satisfied. – All ”central” middle control points (if there are any) are classified asthe basic (free) ones. – If the ”Projections Relation” does not hold then there are n − n − n −
3) ”side” middle control points. – If the ”Projections Relation” holds then there are n − n − n −
3) ”side” middle control points; in every pair ( ˜ L j , ˜ R j ) ( j = 2 , . . . , n −
2) one controlpoint is basic and another one is dependent.Dependencies of the dependent middle control points are described in the the-orem’s proof.
Proof
See Appendix, Section D. n = 4 and n = 5For n = 4 the ”Middle” system consists of two indexed equations ”Eq(2)” and ”Eq(3)” and there are two middle control points ˜ L and ˜ L .Theorem 5 implies that if the ”Projections Relation” is not satisfied, thenthe local MDS is empty. Both middle control points are dependent and theirdependencies on the basic E , V , D , T -type control points are defined by the ”Mid-dle” system which has a 2 × ”Projections Relation” holds then equations ”Eq(2)” and ”Eq(3)” areno longer independent and an additional degree of freedom is available; one ofthe control points ˜ L , ˜ R becomes basic (free).For n = 5, if the ”Projections Relation” does not hold then one of the controlpoints ˜ L , ˜ L , ˜ R , ˜ R is basic (free) and the others are dependent. If the ”Pro-jections Relation” is satisfied then there are two basic (free) control points, onein every pair ( ˜ L , ˜ R ) and ( ˜ L , ˜ R ). (a) (b) (c) Fig. 17: Local templates for the classification of V , E -type control points in caseof global bilinear in-plane parametrisation ˜Π ( bilinear ) .Fig. 18: Possible mesh configuration, which automatically satisfies the ”CircularConstraint” . (a) (b) (c) Fig. 19: Local templates for the classification of D , T -type control points ad-jacent to an inner vertex in case of global bilinear in-plane parametrisation˜Π ( bilinear ) . (a) (b) Fig. 20: Local templates for the classification of D , T -type control points adja-cent to a boundary vertex in case of global bilinear in-plane parametrisation˜Π ( bilinear ) . Fig. 21: An illustration for the ”Projections Relation” . (a) (b) Fig. 22: Local templates for the classification of the middle control points incase of global bilinear in-plane parametrisation ˜Π ( bilinear ) .
12 Global MDS
The main purpose of the current Section is to define the global MDS based onthe local analysis given in Section 11. One should try to ”put together” localtemplates without contradictions. More precisely , the order of construction ofthe local MDS should be defined. It is important to remember, that even if localtemplates do not intersect geometrically, the construction of local MDS is usu-ally based on the assumption that some control points are already classified andit is possible to define dependencies on these control points. Therefore, the or-der of construction of the local MDS plays the principal role in the constructionof the global MDS.Although a ”pure” global MDS (MDS based on G -conditions only) alwaysexists, application of an ”additional” constraint requires construction of some definite suitable local templates, which do not always fit together. If the globalclassification succeeds then it will be said that an instance of global MDS, whichfits the ”additional” constraints, is constructed. Otherwise, according to thecurrent approach, MDS of a higher degree should be considered; an attempt torebuild the local templates defined in Section 11 is never done. n ≥ For n ≥ Algorithm 1:
Algorithm for construction of global MDSin case of a global bilinear in-plane parametrisation ˜Π ( bilinear ) ”Stage 1” For every non-corner mesh vertex, construct a local MDS for theclassification of V , E -type control points (see Subsection 11.1). The choiceof the local template for every vertex is made according to the local meshstructure and should fit the given ”additional” constraint. At the end ofthe stage all V , E -type control points are classified and all ”Eq(0) -typeequations are satisfied. ”Stage 2” For every non-corner mesh vertex, construct a local MDS for clas-sification of D , T -type control points (see Subsection 11.2). The choiceof the local template for every vertex should fit the given ”additional”constraint. At the end of the stage all V , E , D , T -type control points areclassified and all ”Eq(0) and ”Eq(1)” -type equations are satisfied. ”Stage 3” For every inner edge, construct a local MDS for the classificationof the middle control points (see Subsection 11.3). The choice of thelocal template for every edge depends on the local geometrical structureof the mesh. At the end of the stage all control points are classified andall G -continuity equations are satisfied. Lemma 10:
For any n ≥ B ( n ) ( ˜Π ( bilinear ) ) of the global MDS, that fits the ”additional”constraints.Dimensionality of the global MDS is studied in Subsection 12.3. n = 4 D -type control points For n = 4, construction of the global MDS becomes more complicated since thelocal templates defined in Section 11 may intersect. The intersection occursbetween local templates for classification of D , T -type control points adjacent toend-vertices of the same inner edge (see Figure 24). The intersection containsthe single control point - D -type control point of the edge; the problem arisesif both templates define the control point as dependent.In general, Algorithm 1 remains valid for n = 4. Moreover, the only stagewhich should be made more accurate is ”Stage 2” . At this stage for n ≥ D -type control point (if needed) may be madearbitrarily in case of ambiguity. For n = 4 the order in which the vertices aretraversed and the choice of the dependent D -type control points (if needed) playthe principal role.According to the current approach, the global classification of D , T -typeis said to exist if it is possible to ”put together” the local templates withoutcontradiction. In order to do it, one should specify the traversal order of thevertices and the choice of dependent D -type control point (if any) for everytemplate. The classification involves a global analysis and may fail for somekinds of the ”additional” constraints. If the classification succeeds, the globalMDS exists and the rest of the control points are classified precisely as in thecase of n ≥ T B ( D,T ) (seeSubsection 11.2.4), D -type control points can be classified prior to classificationof T -type control points. Furthermore, as soon as a global classification of D -type control points is made, classification of T -type control points can be madelocally, separately for every vertex and can not lead to any conflicts betweendifferent vertices. It implies that global classification of D -type control pointsis the most difficult and the important step in the construction of the globalMDS. The following notations will be used. Definition 18:
A vertex ˜ V is called D -relevant if the local template for classification of D , T -type control points contains a dependent D -type control point.It will be said, that the D -relevant vertex ˜ V uses an adjacent D -typecontrol point ˜ D ( j ) if D ( j ) enters Equation 37 with a non-zero coefficient. A D -type control point is said to be assigned to an adjacent D -relevantvertex if it is classified as dependent according to the local template atthe vertex. The vertex is called ”owner” of the assigned D -type controlpoint.There are two types of D -relevant vertices- Inner even vertices, excluding regular 4-vertices.- Boundary vertices which use template T B ( D,V ) , or in other words, bound-ary vertices with non-colinear adjacent boundary edges in the case of aclamped boundary condition. The following small examples show what kind of difficulties one may encounterduring construction of global MDS of degree 4 which fits a given ”additional”constraint.In the case of a clamped boundary condition, the three meshes presentedin Figure 25 have no sufficient D -type control points in order to assign an”owned” D -type control point to every D -relevant mesh vertex (arrows in theFigure show the ”owner” vertices for D -type control points). Figure 25(a) -the inner 4-vertex has not its own D -type control point. Figure 25(b) - noneof the inner 4-vertices has its own D -type control point because D -points ofthe dashed edges do not contribute to the ”Circular Constraint” for the innervertices. Figure 25(c) - there are no sufficient D -type control points for eitherone of two inner 4-vertices.An additional example of Figure 26 shows the situation when an inner 3-vertex ˜ V (which does not use the adjacent D -type control points itself) also”does not help” to get free D -type control points for the neighboring 4-vertices.This is because the D -type control points of the dashed edges do not contributeto the ”Circular Constraint” for vertices ˜ V (1) , ˜ V (2) and ˜ V (3) . D , T -type control points Classification of D , T -type control points will explicitly be made for any kind of”additional” constraints, excluding the clamped boundary condition. For mostof the mesh configurations, the general algorithms work as well in the case of aclamped boundary condition. Different techniques, which may help in case offailure of the general algorithm, are provided. The classification process heavilyuses different graph-like structures and graph-theory algorithms (see [11]). Thenext Definition introduces some special ”graph-related” notations, which will beused in the current discussion. Definition 19: • Mesh vertex or primary vertex - vertex of the initial mesh. • Secondary vertex - auxiliary (symbolic) vertex in the middle of the edge. • Mesh edge or full edge - edge of the initial mesh. • Half-edge - edge connecting a primary vertex and an adjacent secondaryvertex.First, an undirected D -dependency graph will be constructed. Then the span-ning tree algorithm will be applied to every connected component of the depen-dency graph and a directed D -dependency forest will be built. In most cases,the dependency forest allows to define the traversal order for the vertices andto assign D -type control point to every D -relevant vertex. Construction of an undirected D -dependency graph An undirected D -dependencygraph is constructed according to the following Algorithm. Algorithm 2:
Algorithm for construction of the D -dependency graph (1) Add secondary vertex ˜ S ( ij ) in the middle of every mesh edge ( ˜ V ( i ) , ˜ V ( j ) )(see Figure 27(a)). (2) Delete half-edge ( ˜ V ( i ) , ˜ S ( ij ) ) if vertex ˜ V ( i ) does not use the D -type controlpoint of the mesh edge ( ˜ V ( i ) , ˜ V ( j ) ) (see Definition 18 and Figure 27(b)). (3) Delete a secondary vertex if after elimination of half-edges (step (2) of theAlgorithm) no half-edge connected to it remained. (4)
Delete a primary vertex if it does not use any D -type control point (if itis not D -relevant ).Now the D -dependency graph consists of D -relevant primary vertices andsuch secondary vertices that the correspondent D -type control point is used bysome primary vertex . A primary and a secondary vertex are connected by a half-edge if and only if the primary vertex uses the D -type control point corre-sponding to the secondary vertex . It will be said that two primary vertices ˜ V ( i ) and ˜ V ( j ) are connected by the full edge if both of them use D -type control pointof the mesh edge ( ˜ V ( i ) , ˜ V ( j ) ) (both half-edges ( ˜ V ( i ) , ˜ S ( ij ) ) and ( ˜ S ( ij ) , ˜ V ( j ) ) belongto the D -dependency graph). If ˜ S ij is connected to only one of the primary ver-tices , say ˜ V ( i ) , then the half-edge ( ˜ V ( i ) , ˜ S ( ij ) ) will be called a dangling half-edge .From the topological point of view the graph consists of a few (possible zeroor one) connected components. Each connected component may contain pairsof the primary vertices connected by the full edges and the secondary vertices connected to the primary ones by the dangling half-edges . Construction of directed a D -dependency forest when any connected com-ponent of D -dependency graph has a dangling half-edge Algorithm 3: Algorithm for the construction of the D -dependency forestwhen every connected component of the D -dependency graphhas at least one dangling half-edge For every connected component of D -dependency graph, define a D -dependencytree in the following way. (1) Choose a primary vertex which has a dangling half-edge (at least one suchvertex exists for every connected component), denote this vertex by ˜ R . (2) Build a spanning tree of the primary vertices of the connected componentusing the full edges only (this is possible because the component obviouslyremains connected with respect to the primary vertices after eliminationof all dangling half-edges (see Figure 28(b)). (3)
Define vertex ˜ R to be the root of the spanning tree. (4) Direct every mesh edge of the tree from the upper vertex to a lower one(according to the hierarchy of the spanning tree, see Figure 28(c)). (5)
Direct the mesh edge corresponding to a dangling half-edge at ˜ R towards˜ R . From this moment this mesh edge is defined as belonging to the D -dependency tree of ˜ R (the correctness of this definition is shown in Lemma11). Note 5:
The D -dependency forest consists of the primary D -relevant verticesand directed mesh edges ; at this step one may forget about auxiliary secondaryvertices and half-edges. Note 6:
Assigning directions to the mesh edges which correspond to the danglinghalf-edges of the root vertices may lead to the situation when D -dependencytrees of different components of the D -dependency graph become connected bythese directed edges (see Figure 29). Such trees still will be referred to as thedifferent trees of the D -dependency forest. The mesh vertex and its adjacentdirected mesh edge may belong to different D -dependency trees; a directedmesh edge always belongs to the D -dependency tree of the vertex it points to. Lemma 11:
The directions assigned to mesh edges as a result of the constructionof D -dependency forest, obey the following rules (1) Every mesh edge is either undirected or its direction is correctly defined.In particular, it implies that every mesh edge belongs to at most one D -dependency tree. (2) Every D -relevant mesh vertex uses D -type control points of either undi-rected mesh edges or of the directed mesh edges that belong to the same D -dependency tree as the vertex itself. (3) Every D -relevant mesh vertex ˜ V uses exactly one D -type control pointwhich belongs to the directed edge pointing towards ˜ V . This D -typecontrol point is denoted ˜ D ( ˜ V ) (see Figure 28(c)). Proof
See Appendix, Section D. Explicit classification of D , T -type control points when any connected com-ponent of D -dependency graph has a dangling half-edge As soon as the D -dependency forest is constructed, the global classification of D , T -type controlpoints can be made according to Algorithm 4. The correctness of classificationis justified by Lemma 11. In particular, Lemma 11 implies that classificationof D -type control points for the different trees of the D -dependency forest canbe made independently, without any conflicts. Algorithm 4:
Algorithm for the classification of D , T -type control pointsbased on D -dependency forest ”Step 1”(1) For every D -relevant boundary vertex, classify T -type control points asbasic according to template T B ( D,V ) . (2) Classify as basic D -type control points of all mesh edges which remainedundirected after construction of the D -dependency forest. (3) For every tree in the D -dependency forest do the following • Assign the D -type control point of every directed mesh edge to the mesh vertex this edge points to. • Traverse the D -dependency tree down-up, level by level. • At every level for every D -relevant vertex ˜ V (the order of verticesbelonging to the same level is not important) choose the assigned D -type control point ˜ D ( ˜ V ) as dependent in the local template of the vertex.Define the dependency of the corresponding Z -component D ( ˜ V ) accordingto the local template (Equation 44 for an inner vertex and Equation 48for a boundary vertex). D ( ˜ V ) may depend on- V -type and E -type control points which are classified as basic duringthe first stage of the classification process.- D -type control points which are classified as basic during traversingthe lower levels of the tree.- T -type basic fixed control points adjacent to D -relevant boundaryvertices. ”Step 2” For every non-corner vertex (excluding D -relevant boundary vertices),classify T -type control points locally according to the local template (seeSubsections 11.2.3 and 11.2.4).For example, in the mesh fragment shown in Figure 30, ˜ V is a leaf-vertex of some D -dependency tree, ˜ V (4) is the ”father” of ˜ V in the D -dependency tree and ˜ V (1) is the root vertex of some other D -dependency tree. Then ˜ D (2) , ˜ D (3) , ˜ D (5) , ˜ D (6) are basic D -type control points. ˜ D (1) is a dependent control point, however itdoes not contribute to the ”Circular Constraint” for ˜ V since edges numbered 2 and 6 are colinear.This ”Circular Constraint” defines the dependency of D (4) = D ( ˜ V ), D (2) , D (3) , D (5) (and E and V -type basic control points) according to theequation (cid:80) j =2 (cid:104) ˜ e ( j − , ˜ e ( j +1) (cid:105)(cid:104) ˜ e ( j − , ˜ e ( j ) (cid:105)(cid:104) ˜ e ( j ) , ˜ e ( j +1) (cid:105) ( D ( j ) − E ( j ) + V ) = 0 (51) Absence of D -relevant boundary vertices as a sufficient condition for globalclassification of D , T -type control points Theorem 6 presents a sufficientcondition for the existence of global classification of D , T -type control points.It is important to pay attention to the fact that the presence of inner D -relevant vertices depends on the mesh structure, while the presence of boundary D -relevant vertices depends both on mesh structure and on the type of applied”additional” constraints. Theorem 6:
Absence of D -relevant boundary vertices implies that the D -dependencygraph is either empty or every connected component of D -dependency graphcontains at least one dangling half-edge. Therefore the global classification of D , T -type control points exists and can be made by successive application ofAlgorithms 2, 3 and 4. Proof
See Appendix, Section D.
Case when there exist D -relevant boundary vertices In most of practical sit-uations, even in the presence of D -relevant boundary vertices, every connectedcomponent of the D -dependency graph has at least one dangling half-edge dueto the neighborhood of inner odd vertices or regular 4-vertices. In this case theclassification of D , T -type control points exists and is made according to Algo-rithms 2, 3 and 4. If some connected component has no dangling half-edges,then Algorithm 3 fails to construct the D -dependency tree for this component.Algorithm 5 (see Appendix, Section B) presents a simple modification ofAlgorithms 3 and 4 which may help to classify D , T -type control points forthose components of the D -dependency graph, which have no dangling half-edges. The construction is made separately for every such component and doesnot affect the construction of the classification solution for components withdangling half-edges.The principal reason of failure of Algorithm 3 for a connected component˜ C which has no dangling half-edges is an impossibility to assign D -type controlpoint to the root vertex of the spanning tree. Algorithm 5 tries to overcomethis problem by splitting a full edge of a component so that the edge does notparticipate in the spanning tree of ˜ C . Algorithm 5 does not necessarily succeed.If the Algorithm fails (for one of the reasons which are explained below), anyone of the following possibilities may be chosen. • Pass to MDS of degree 5 which is well defined for any kind of ”additional”constraints. • Pass to the solution which combines patches of degree 5 along the bound-ary and patches of degree 4 in the inner part of the mesh (see Part VI). • Try to create a dangling half-edge by a local change of the initial mesh,for example by the regularisation of one of the inner 4-vertices, as shownin Figure 31.
Algorithm 5:
See Appendix, Section B.
The results of Subsection 12.2.3 lead to the following conclusions.
Lemma 12: (1)
Global ”pure”
MDS ˜ B (4) ( ˜Π ( bilinear ) ) (MDS which relates to G continuityconstraints alone) is well defined for any mesh configuration. (2) For any ”additional” constraint which does not involve D -relevant bound-ary mesh vertices, an instance of ˜ B (4) ( ˜Π ( bilinear ) ) , which fits the ”addi-tional” constraint, exists for any mesh configuration. In particular, asuitable ˜ B (4) ( ˜Π ( bilinear ) ) always exists for any kind of interpolation andfor simply-supported boundary conditions.The MDS is constructed according to Algorithm 1, where ”Stage 2” is accom-plished by successive application of Algorithms 2, 3 and 4. The dimensionalityof ˜ B (4) ( ˜Π ( bilinear ) ) is analysed in Subsection 12.3.For any ”additional” constraint which involve D -relevant boundary ver-tices (ex. clamped boundary condition for vertices with non-colinear adjacentboundary edges), the current approach may fail to construct an instance of˜ B (4) ( ˜Π ( bilinear ) ) which fits this ”additional” constraint. Nevertheless even inthe presence of D -relevant boundary vertices, the general classification algo-rithm or its modification (Algorithm 5) will work for most mesh configurations.A solution which combines patches of degrees 4 and 5 is presented in Part VI. Theorem 7:
For a global bilinear in-plane parametrisation ˜Π ( bilinear ) and forany n ≥ B ( n ) G (subset of MDS which participates in the G -continuity condition) and the dimension of ˜ B ( n ) (full dimension of MDS) aregiven by the following formulas (see Definition 10 and Lemma 3) | ˜ B ( n ) G | = 3 | V ert non − corner | + | V ert boundarynon − corner | + (2 n − | Edge inner | + | V ert inner − regular | + | Edge inner, ” P rojections Relation ” holds | (52) | ˜ B ( n ) | = | ˜ B ( n ) G | + | ˜ CP ( n ) F | = | ˜ B ( n ) G | + ( n − | F ace inner | + ( n − n − | F ace boundarynon − corner | +( n − | F ace corner | (53) Proof
See Appendix, Section D.The examples given in Figures 23, 32, 24 provide illustrations to theformulas given in Equation 52. The Figures present instances of ˜ B (5) G (Figure23) and ˜ B (4) G (Figures 32 and 24) for the same mesh. The mesh contains oneregular 4-vertex ˜ V ( reg ) and the ”Projections Relation” holds for edge ˜ e (” P R ”) .Control points which belong to ˜ B ( n ) G are marked by filled circles of different colors(orange, blue, green, violet and pink for V , E , D , T -type and middle control pointsrespectively). Arrows in Figures 32 and 24 show which D -type control pointsare assigned to D -relevant vertices.Figure 23 shows a ”pure” instance of ˜ B (5) G (no ”additional” constraints areapplied). ˜ B (5) G contains 67 control points (among 117 control points of ˜ CP (5) G ),which precisely fits Equation 52. Figures 32 and 24 respectively show a ”pure”instance of ˜ B (4) G and an instance which fits the clamped boundary condition.Both instances have the same dimension and contain 47 control points.An additional example, which allows to verify the correctness of Equation53, is given in Section 18. )reg( V~ )"PR(" e~ Fig. 23: An example of a ”pure” global MDS ˜ B (5) ( ˜Π ( bilinear ) . Fig. 24: An example of a global MDS ˜ B (4) ( ˜Π ( bilinear ) , which fits the clampedboundary conditions. (a) (b) (c) Fig. 25: Examples of such meshes that not every D -relevant mesh vertex has itsown D -type control point.Fig. 26: An example of mesh configuration when 3-vertex does not contributeto the ”Circular Constraint” for adjacent 4-vertices. (a) (b) Fig. 27: An illustrations for the construction of the D -dependency graph. (a) (b) (c) Fig. 28: Construction of D -dependency tree for a connected component of D -dependencygraph. (a) Connected component of D -dependency graph (here the structure of thecomponent is not correct in the meaning that it does not correspond to any planar mesh,the Figure serves only as an illustration for Algorithm 3). (b) Spanning tree of theconnected component. (c) D -dependency tree of the connected component. (a) (b) (c) Fig. 29: An example of two different D -dependency trees connected by the directed edgesat the root vertices. (a) Planar mesh vertices and edges. (b) D -dependency graphconsisting of two connected components. (c) D -dependency trees for two components of D -dependency graph. Fig. 30: An example of the classification of D -type control points.Fig. 31: Regularization of an inner 4-vertex. Here ˜ V ( init ) is a vertex of the initial meshand ˜ V ( reg ) is the corresponding regularized vertex. In D -dependency graph, ˜ V ( reg ) helpsto obtain dangling half-edges for connected components of the adjacent vertices.Fig. 32: An example of a ”pure” global MDS ˜ B (4) ( ˜Π ( bilinear ) . Part V. MDS for a quadrilateral mesh witha smooth global boundary
To construct higher order approximations/interpolations, one needs also to han-dle smooth boundaries without reducing them to polygonal lines! This chapterdeals with such constructions : planar meshes with a smooth global boundary(like the mesh shown in Figure 33).The bilinear in-plane parametrisation is no longer sufficient at the boundary.However, the global in-plane parametrisation is constructed in such a manner,that the local templates should be changed only for the boundary vertices andfor inner edges adjacent to the boundary. Therefore, a study of the continuityconstraints for the edges adjacent to the boundary plays the principal role inconstruction of an MDS.Like in the case of a polygonal global boundary, a ”pure” ˜ B ( n ) is constructedfor any n ≥ D -relevant, a relatively high degreeof the weight functions for the inner edges adjacent to the boundary may resultin the failure of ˜ B (4) construction for some ”additional” constraints.Fig. 33: An example of a planar domain with a smooth global boundary.
13 Definitions,Mesh limitations and In-Planeparametrisation
Let us recall some definitions from ... ˜ A, ˜ B, ˜ C, ˜ D - four vertices of a boundary mesh element, where ˜ D and ˜ C are theboundary vertices (see Figure 34(a)).˜ E , ˜ F - two inner control points of the cubic boundary curve (see Figure 34(a)). Two adjacent boundary elements ˜ λ, ˜ λ (cid:48) , ˜ γ, ˜ γ (cid:48) , ˜ ρ, ˜ ρ (cid:48) - vertices of two adjacent boundary mesh elements (see Figure35)˜ T (cid:48) λ , ˜ T (cid:48) ρ - control points of the boundary curves adjacent to the vertex ˜ γ (cid:48) (seeFigure 35).The notion of planar quadrilateral mesh is slightly generalised and it is as-sumed that • The geometry of every inner edge is described by a straight segment. • The edges along the global boundary of the planar domain have a cubicparametric B´ezier representation and the concatenation between any pairof adjacent edges is (non-degenerated) G -smooth.All basic assumptions listed in Section 8 remain valid in the current case.Moreover, for simplicity, the mesh is assumed to have no corner elements, or inother words, every boundary vertex is supposed to have exactly one adjacentinner edge. There is no reason to change the bilinear type of parametrisation for inner meshelements. For a boundary mesh element, we introduce a bicubic parametrisation˜ P ( u, v ) = (cid:80) i,j =0 ˜ P ij B ij in order to fit the boundary curve data. The followingprinciples define the choice of parametrisation for a boundary mesh element. • In order to make the definition of a global in-plane parametrisation possible,it should be linear along the edge with two inner vertices and the parametri-sations of two boundary elements should agree along edges with a commonboundary vertex. • The parametrisation should have a minimal possible influence on G -continuityequations for edges with two inner vertices. It makes it possible to reuse thelocal templates constructed in the case of global bilinear in-plane parametri-sation. • Partial derivatives of the parametrisation along the edges with one boundaryvertex should have the minimal possible degrees. It allows decreasing thenumber of linear equations which are sufficient in order to guarantee thatthe G -continuity condition is satisfied.Let the upper edge of a boundary mesh element lie on the global boundaryof the planar domain. The following choice of the control points for in-planeparametrisation of the element is adopted (see Figure 34(b)). • ˜ P , ˜ P , ˜ P , ˜ P and ˜ P , ˜ P are given respectively by the vertices of theplanar element and by the control points of the cubic boundary curve. • ˜ P , ˜ P , ˜ P , ˜ P , ˜ P , ˜ P are given by the degree elevation of the bilinearparametrisation up to degree 3. This allows the linearisation of the G -continuity for the lower edge exactly as it was done in the case of a globalbilinear in-plane parametrisation. • ˜ P , ˜ P are also given by the degree elevation of the bilinear parametrisation.It leads to the minimal possible degrees of the partial derivatives of in-planeparametrisation in directions along the left and the right edges. • ˜ P , ˜ P are chosen in such a manner, that the partial derivatives of in-planeparametrisation in the cross direction for the left and the right edges have theminimal possible formal degrees (degree 2). Here it is assumed that prior tothe choice of ˜ P and ˜ P all other control points of in-plane parametrisationare already fixed according to the three first items.The explicit formulas for the control points of in-plane bicubic parametrisationof a boundary mesh element are given in Technical Lemma 2 (see Appendix,Section C). Technical Lemma 3 (see Appendix, Section C) describes a sufficient geomet-rical condition for the regularity of the chosen in-plane parametrisation for theboundary mesh elements. A study of the regularity in the current work is re-stricted to this simple condition which is s quite natural , meaning that it issatisfied for the ”non-degenerated” structures of the boundary elements. Ofcourse, a more detailed analysis of the regularity conditions can be made. ˜Π ( bicubic ) For every boundary element, let a bicubic in-plane parametrisation be con-structed according to Technical Lemma 2 (see Appendix, Section C) and let it be regular (the requirements of Technical Lemma 3 (see Appendix, Section C)are satisfied). Then according to Definition 2, Paragraph 1.3.2, the collectionof bicubic parametrisations for boundary mesh elements and bilinear parametri-sations for inner mesh elements defines a global regular in-plane parametrisation˜Π ( bicubic ) ∈ ˜ PAR (3) . Clearly, degree 3 is just a formal degree of the parametri-sation; all inner mesh elements are of degree 1 and parametrisations of theboundary mesh element have actual degree 3 only along the global boundary. (a) (b)
Fig. 34: A planar mesh element, control points of a boundary curve and control pointsof in-plane parametrisation in the case of a mesh with a smooth global boundary.
14 Conventional weight functions14.1 Weight functions for an edge with two inner vertices
Bilinear parametrisation of the inner mesh elements and a special choice of thecontrol points for in-plane parametrisation of the boundary mesh elements leadto the following Lemma.
Lemma 13:
Let a mesh edge have two inner vertices. Then the conventionalweight functions defined according to the global in-plane parametrisations ˜Π ( bicubic ) and ˜Π ( bilinear ) , coincide. Let ˜ P be the restriction of a global in-plane parametrisation ˜Π ( bicubic ) on aboundary mesh element. For the edges with one boundary vertex, the first-order partial derivatives of ˜ P have degree 0 in the direction along the edgesand formal degree 2 in the cross direction. The explicit formulas of the partialderivatives are given in Technical Lemma 4 (see Appendix, Section C).Let further ˜ L ( u, v ) and ˜ R ( u, v ) be the restrictions of the global in-planeparametrisation ˜Π ( bicubic ) on the adjacent boundary elements. Although theformal degree of ˜ L ( u, v ) and ˜ R ( u, v ) is m = 3, none of the partial deriva-tives ˜ L u , ˜ R u , ˜ L v has the full actual degree. Therefore, the linear system of N umEqF ormal = n + 2 m = n + 6 equations, described in Lemma 2,paragraph5.1 ,contains redundant equations. An analysis of the conventional weight func-tions allows to decrease the number of sufficient linear equations and finally tocompute precisely the rank of the corresponding linear system. The polynomialrepresentations of the partial derivatives ˜ L u , ˜ R u , ˜ L v in both B´ezier and powerbasis play a very important role in the study of the weight functions. Lemma 14:
Let ˜ L ( u, v ) and ˜ R ( u, v ) be the restrictions of global in-plane parametri-sation ˜Π ( bicubic ) on two adjacent boundary mesh elements. Then the first-orderpartial derivatives of the in-plane parametrisations along the common edge havethe polynomial representations that follows in terms of the initial mesh data.Here ˜ λ, ˜ λ (cid:48) , ˜ γ, ˜ γ (cid:48) , ˜ ρ, ˜ ρ (cid:48) and ˜ T (cid:48) λ and ˜ T (cid:48) ρ denote respectively vertices of the elementsand control points of the boundary curves (see Subsection 13.1.1 and 2.4 andFigure 35 for the definitions). • ˜ L v = ˜ γ (cid:48) − ˜ γ is a constant (polynomial of degree 0). • ˜ L u , ˜ R u are polynomials of formal degree 2. The coefficients of ˜ L u and ˜ R u with respect to B´ezier and power basis are given below (cf section 2.4):˜ λ = ˜ γ − ˜ λ ˜ λ ( power )0 = ˜ λ = ˜ γ − ˜ λ ˜ λ = [(˜ γ − ˜ λ )+(˜ γ (cid:48) − ˜ λ (cid:48) )] ˜ λ ( power )1 = 2(˜ λ − ˜ λ ) = (˜ γ (cid:48) − ˜ γ ) − (˜ λ (cid:48) − ˜ λ )˜ λ = 3(˜ γ (cid:48) − ˜ T (cid:48) λ ) ˜ λ ( power )2 = ˜ λ − λ +˜ λ = 3(˜ γ (cid:48) − ˜ T (cid:48) λ ) − (˜ γ (cid:48) − ˜ λ (cid:48) )˜ ρ = ˜ ρ − ˜ γ ˜ ρ ( power )0 = ˜ ρ = ˜ ρ − ˜ γ ˜ ρ = [(˜ ρ − ˜ γ )+(˜ ρ (cid:48) − ˜ γ (cid:48) )] ˜ ρ ( power )1 = 2(˜ ρ − ˜ ρ ) = (˜ ρ (cid:48) − ˜ ρ ) − (˜ γ (cid:48) − ˜ γ )˜ ρ = 3( ˜ T (cid:48) ρ − ˜ γ (cid:48) ) ˜ ρ ( power )2 = ˜ ρ − ρ + ˜ ρ = 3( ˜ T (cid:48) ρ − ˜ γ (cid:48) ) − (˜ ρ (cid:48) − ˜ γ (cid:48) ) (54)Vectors ˜ λ i , ˜ ρ i ( i = 0 , . . . ,
2) are shown in Figure 36(a). Figure 36(b) showsvectors ˜ L v , − ˜ λ ( power )0 , − ˜ λ ( power )1 + ˜ L v , ˜ λ ( power )2 , ˜ ρ ( power )0 , ˜ ρ ( power )1 + ˜ L v , ˜ ρ ( power )2 .These vectors define the actual degrees of the weight functions and minimalnumber of linear equations which are sufficient in order to satisfy the G -continuity condition. Lemma 15 describes relations between coefficients of the weight functions andcoefficients of the partial derivatives ˜ L u , ˜ R u , ˜ L v . Explicit formulas for coeffi-cients of the weight functions in terms of the initial mesh data with respect tothe B´ezier and the power basis immediately follow from Lemmas 14 and 15.Although the explicit formulas are useful in the following analysis, they becomerelatively complicated in the case of bicubic parametrisation and are given inTechnical Lemmas 5 and 6 (see Appendix, Section C). Lemma 15:
Let L ( u, v ), R ( u, v ) be the restrictions of the global in-plane parametri-sation ˜Π ( bicubic ) on two adjacent boundary mesh elements. Then for the commonedge of the elements, conventional weight functions c ( v ), l ( v ) and r ( v ) have for-mal degrees 4, 2 and 4 respectively. Relations between the coefficients of theweight functions with respect to the B´ezier and power bases and coefficients ofthe partial derivatives ˜ L u , ˜ R u , ˜ L v are given below. l i = (cid:104) ˜ ρ i , ˜ γ (cid:48) − ˜ γ (cid:105) l ( power ) i = (cid:104) ˜ ρ ( power ) i , ˜ γ (cid:48) − ˜ γ (cid:105) r i = −(cid:104) ˜ λ i , ˜ γ (cid:48) − ˜ γ (cid:105) r ( power ) i = −(cid:104) ˜ λ ( power ) i , ˜ γ (cid:48) − ˜ γ (cid:105) (55) c k = (cid:18) (cid:19) (cid:80) i + j = k ≤ i, j ≤ (cid:16) (cid:17) (cid:16) (cid:17) (cid:104) ˜ λ i , ˜ ρ j (cid:105) c ( power ) k = (cid:80) i + j = k ≤ i, j ≤ (cid:104) ˜ λ ( power ) i , ˜ ρ ( power ) j (cid:105) (56)Here i = 0 , . . . , l ( v ) and r ( v ) and k = 0 , . . . , c ( v ). Note 7:
From Lemma 15 and according to the geometrical meaning of ˜ λ , ˜ λ ,˜ ρ , ˜ ρ (see Lemma 14) it follows that l = l ( power )0 (cid:54) = 0 , l (cid:54) = 0 r = r ( power )0 (cid:54) = 0 , r (cid:54) = 0 (57) c = 0 (58) A very important relation between the actual degrees of the weight functions isgiven in Lemma 16 (see Subsection 2.5 for definitions).
Lemma 16:
Let the conventional weight functions c ( v ), l ( v ), r ( v ) for the com-mon edge of two adjacent boundary mesh elements be defined according to theglobal in-plane parametrisation ˜Π ( bicubic ) . Then the actual degrees of the weightfunctions are connected by the following relation: deg ( c ) ≤ max deg ( l, r ) + 2 (59) Proof
See Appendix, Section D.
The different geometrical configurations of two adjacent boundary elements re-sult in the actual different degrees of the weight functions. Examples shownin Figure 37 provide some intuition regarding the geometrical meaning of theactual degrees of the weight functions.(a) Example of degrees (4 , ,
2) for the weight functions c, l, r . Vectors ˜ λ ( power )2 and ˜ ρ ( power )2 are not parallel to each other and both of them are not parallelto the vector ˜ γ (cid:48) − ˜ γ .(b) Example of degrees (3 , ,
2) for the weight functions c, l, r . Vectors ˜ λ ( power )2 and ˜ ρ ( power )2 are parallel to each other but they are not parallel to thevector ˜ γ (cid:48) − ˜ γ . deg ( c ) ≤ if and only if ˜ λ ( power )2 and ˜ ρ ( power )2 are parallel.(c) Example of degrees (3 , ,
1) for the weight functions c, l, r . Three vectors˜ λ ( power )2 , ˜ ρ ( power )2 and ˜ γ (cid:48) − ˜ γ are parallel. max deg ( l, r ) ≤ if and only if vectors ˜ λ ( power )2 and ˜ ρ ( power )2 are parallel to ˜ γ (cid:48) − ˜ γ .(d) Example of degrees (2 , ,
1) for the weight functions c, l, r . Vectors ˜ λ ( power )2 ,˜ ρ ( power )2 and ˜ γ (cid:48) − ˜ γ are parallel. Equality ˜ λ ( power )1 = ˜ ρ ( power )1 implies that deg ( c ) ≤ , ,
0) for the weight functions c, l, r . Vectors ˜ λ ( power )2 ,˜ ρ ( power )2 , ˜ λ ( power )1 , ˜ ρ ( power )1 and ˜ γ (cid:48) − ˜ γ are parallel. l ( power )1 = r ( power )1 = 0 if and only if vectors ˜ λ ( power )2 , ˜ ρ ( power )2 , ˜ λ ( power )1 , ˜ ρ ( power )1 are parallel to˜ γ (cid:48) − ˜ γ .The examples show that different geometrical configurations lead to a varietyof the triples ( deg ( c ) , deg ( l ) , deg ( r )). The actual degree of the weight function c is not uniquely defined by the degrees of the weight functions l and r . Fig. 35: Two adjacent boundary mesh elements in the case of a mesh with asmooth global boundary. (a) (b)
Fig. 36: Coefficients of ˜ L u , ˜ R u , ˜ L v with respect to B´ezier and to power bases for twoadjacent mesh elements in the case of global in-plane parametrisation ˜Π ( bicubic ) . G -continuity conditions (a) (4,2,2) (b) (3,2,2) (c) (3,1,1)(d) (2,1,1) (e) (2,0,0) Fig. 37: Different geometrical configurations of two adjacent boundary meshelements lead to different actual degrees of the conventional weight functions.
15 Linear form of G -continuity conditions15.1 G -continuity conditions for an edge with two innervertices Lemma 13 clearly implies that the following Lemma holds.
Lemma 17:
For any edge with two inner vertices, the linear system of G -continuity equations in the case of global in-plane parametrisation ˜Π ( bicubic ) remains unchanged with respect to the case of bilinear global in-plane parametri-sation ˜Π ( bilinear ) . G -continuity conditions G -continuity conditions for an edge with one boundaryvertex In the case of global in-plane parametrisation ˜Π ( bicubic ) , the conventional weightfunctions c ( v ), l ( v ), r ( v ) have formal degrees 4, 2, 2 for an edge with one bound-ary vertex (see Lemma 15). Therefore the linearized G -continuity equation(Theorem 3, Equation 17) has the following form n (cid:80) i =0 ∆ L i B ni (cid:80) j =0 l j B j + n (cid:80) i =0 ∆ R i B ni (cid:80) j =0 r j B j + n − (cid:80) i =0 ∆ C i B ni (cid:80) j =0 c j B j = 0 (60)Unlike the case of global bilinear in-plane parametrisation ˜Π ( bilinear ) , not allsummands of the last equation have the same formal degrees: the degree ofthe two first summands is n + 2 while the degree of the last one is n + 3. Ofcourse, this difficulty can be easily overcome by application of the standarddegree elevation to the first two summands. On the other hand it is importantto remember, that n + 2 and n + 3 are just the formal degrees and that the actual degrees of the summands may coincide. For example in the case whenthe actual degrees of the weight functions c, l, r are equal to (3 , ,
2) the degreeelevation is unnecessary, its application will lead to a redundant equation in theresulting linear system.Generally, there are two different approaches to the analysis of Equation60. In the first one, all possible triples of the actual degrees of the weight func-tions should be considered separately. This approach ”starts from geometry”and leads to the different algebraic systems, which correspond to the differentgeometric configurations. The second approach is a much more formal alge-braic one. It starts by degree elevation, and then analyses the unique system ofequations with the aid of the algebraic tools. This analysis eventually leads todifferent sub-cases, which of course correspond to different geometrical configu-rations.Like in the case of global bilinear in-plane parametrisation, the second ap-proach is adopted in the current work. However, while performing an algebraicanalysis, one always should be aware of the alternative way which allows toverify the correctness of the results. For example in the case of degrees (3 , , n + 3 lead to the following Lemma. Lemma 18:
In the case of global in-plane parametrisation ˜Π ( bicubic ) , the systemof the following n + 4 linear equations is sufficient in order to guarantee a G -smooth concatenation between two boundary patches (cid:80) ≤ i ≤ n ≤ j ≤ i + j = s (∆ L i l j + ∆ R i r j ) (cid:16) ni (cid:17) (cid:16) (cid:17) + (cid:80) ≤ i ≤ n ≤ j ≤ i + j = s − (∆ L i l j + ∆ R i r j ) (cid:16) ni (cid:17) (cid:16) (cid:17) + (cid:80) ≤ i ≤ n − ≤ j ≤ i + j = s ∆ C i c j (cid:16) n-1i (cid:17) (cid:16) (cid:17) = 0 (61) G -continuity conditions where s = 0 , . . . , n + 3. A linear system of equations, which is equivalent to the system given in Lemma18 and has a clearer and more intuitive structure, will be constructed in order tosimplify the analysis of MDS. The following notations lead to a more compactform of the linear equations.
Definition 20:
Let sumLR s = (cid:80) ≤ i ≤ n ≤ j ≤ i + j = s (∆ L i l j + ∆ R i r j ) (cid:16) ni (cid:17) (cid:16) (cid:17) for s = 0 , . . . , n + 2 sumLR − = sumLR n +3 = 0 ,sumC s = (cid:80) ≤ i ≤ n − ≤ j ≤ i + j = s ∆ C i c j (cid:16) n-1i (cid:17) (cid:16) (cid:17) for s = 0 , . . . , n + 3 (62)and” Eq ( s )” sumLR s = ( − s +1 s (cid:80) k =0 ( − k sumC k ” sumC − equation ” n +3 (cid:80) k =0 ( − k sumC k = 0 (63)(An expression in the left side of the last equation will also be called ”sumC-equation” ). Lemma 19:
The following linear system of n + 4 equations is sufficient in orderto guarantee a G -smooth concatenation between two boundary patches in thecase of global in plane parametrisation ˜Π ( bicubic ) (cid:26) ” Eq ( s )” s = 0 , . . . , n + 2” sumC − equation ” (64) Proof
See Appendix, Section D.Note, that ”sumC-equation” might be formally written as ”Eq(n+3)” , whichwould lead to a homogeneous system of equations. It was decided to separatethis equation because of its role in the construction of MDS and because of itsspecial properties, some of which are listed in the next Subsection. ”sumC-equation”
Unlike the indexed equations, ”sumC-equation” deals only with the ”central”control points C j ( j = 0 , . . . , n ), none of the ”side” control points ( L j , R j )( j = 0 , . . . , n ) contributes to it.An another important property of ”sumC-equation” is given in Lemma 20. Lemma 20:
Let equation ”C-equation” be defined as follows” C − equation ” (cid:80) n − i =0 ( − i (cid:16) n-1i (cid:17) ∆ C i = − (cid:80) ni =0 ( − i (cid:16) ni (cid:17) C i = 0 (65)(An expression in the left side of the equation will also be called ”C-equation” ).Then ”sumC-equation” can be represented as” C − equation ” c ( power )4 = 0 (66)Lemma 20 in particular means that ”sumC-equation” is automatically satisfiedwhen deg ( c ) ≤
3. Moreover, it shows that if deg ( c ) = 4 then values of C j ( j = 0 , . . . , n ) should necessarily satisfy the ”C-equation” .In addition, Equation 65 implies that ”sumC-equation” and ”C-equation” have a ”global nature” in the meaning that every one of C j control points( j = 0 , . . . , n ) participates in these equations with a non-zero coefficient.
16 Local MDS
In the following discussion, it always will be assumed that edge with one bound-ary vertex is parametrized in such a manner, that ¯ C is the inner vertex ofthe edge and ¯ C n is the boundary vertex. It implies, for example, that equa-tions ”Eq(0)”,”Eq(1)” relate to the inner vertex and equations ”Eq(n+1)”,”Eq(n+2)” to the boundary vertex of the edge (see Figure 39(b)). Lemma 21:
Consider global ˜Π ( bicubic ) parametrisation. Then for any inner ver-tex (1) ”Eq(0)” -type equation remains unchanged with respect to the case ofglobal bilinear in-plane parametrisation ˜Π ( bilinear ) . (2) The couple of ”Eq(0)” -type and ”Eq(1)” -type equations is equivalent tothe couple of the corresponding equations in the case of global bilinearin-plane parametrisation ˜Π ( bilinear ) . Proof
See Appendix, Section D.Lemma 21 leads to the following Conclusions
Conclusion 1:
Consider a global in-plane parametrisation ˜Π ( bicubic ) . Then forany inner vertex, the local templates for classification of V , E , D , T -type controlpoints remain unchanged with respect to the case of global bilinear in-planeparametrisation ˜Π ( bilinear ) .The templates are constructed precisely as described in Subsections 11.1.3and 11.2.3 and are responsible for the satisfaction of ”Eq(0)” -type and ”Eq(1)” -type equations at the inner vertex. Conclusion 2:
All theoretical results related to the Vertex Enclosure problem(see Section 11.2.2) remain valid in the case of global in-plane parametrisation˜Π ( bicubic ) . ”Eq(n+2)” and ”Eq(n+1)” for an edgewith one boundary vertex In the case of a global in-plane parametrisation˜Π ( bicubic ) , equations ”Eq(n+2)” and ”Eq(n+1)” have the following representa-tions for an edge with one boundary vertex. ”Eq(n+2)” equation.∆ L n l + ∆ R n r = 0 (67)Geometrically ”Eq(n+2)” means that the control points ¯ L n , ¯ C n , ¯ R n are colinear.Equation ”Eq(n+2)” never involves C n − ( E -type control point of the edgeadjacent to the boundary vertex). ”Eq(n+1)” equation. n (∆ L n − l + ∆ R n − r ) + 2(∆ L n l + ∆ R n r ) + 4∆ C n − c = 0 (68)This equation involves only control points lying on the mesh boundary ( L n , C n , R n )or adjacent to it ( L n − , C n − , R n − ). The control point C n − ( D -type controlpoint of the edge adjacent to the boundary vertex) does not participate in thisequation. Local templates and different types of ”additional” constraints
The choiceof the local templates for a boundary vertex ˜ V in the case of global in-planeparametrisation ˜Π ( bicubic ) is based on the following two principles. • The template should include all control points adjacent to ˜ V and partici-pating in specified ”additional” constraints. • The template is responsible for the satisfaction of some sub-system ofthe indexed equations (the sub-system contains either the last indexedequation or the last pair of the indexed equation). • The template should contain a minimal possible number of the controlpoints in order to provide additional degrees of freedom to the ”Middle”system, which becomes relatively complicated in the case of ˜Π ( bicubic ) foredges with one boundary vertex.The following two local templates are defined: T B ( bicubic ) (Figure 38(a)). The template includes control points lying atthe global boundary ( V -type control point and two boundary E -type con-trol points). The template is responsible for the satisfaction of equation ”Eq(n+2)” and ”additional” constraints, provided the ”additional” con-straints involve only control points lying at the global boundary. A local MDS contains ˜ V and one of E -type control points ( ˜ E (1) ). Controlpoint ˜ E (3) is dependent and dependency of E (3) is defined according toEquation 67. T B ( bicubic ) (Figure 38(b)). The template includes the control points lying atthe global boundary and adjacent to it ( V , E and T -type control points).The template is responsible for the satisfaction of equations ”Eq(n+2)” , ”Eq(n+1)” and for any type of considered ”additional” constraints.A local MDS contains ˜ V , one of E -type control points ( ˜ E (1) ) and one of T -type control points ( ˜ T (1) ). Control points ˜ E (3) and ˜ T (2) are dependent.First, dependency of E (3) is defined according to Equation 67 and thendependency of T (2) is defined according to Equation 68.The template T B ( bicubic ) is used when the ”additional” constraints involve onlycontrol points lying at the global boundary and degree 4 MDS is considered. Inall other cases template T B ( bicubic ) is used. middle control points for aseparate edge Lemma 17 implies that the following Lemma holds.
Lemma 22:
Consider a global in-plane parametrisation ˜Π ( bicubic ) and let for anyedge with two inner vertices the ”Middle” system of equations and the middle set of the control points be defined according to Definition 16.Then for an edge with two boundary vertices, the local templates responsiblefor the classification of the middle control points are the same as for the globalbilinear in-plane parametrisation ˜Π ( bilinear ) (see Subsection 11.3.1). ”Middle” system of equations and middle control points foran edge with one boundary vertex The following notations are introducedin order to unify the description of the local MDS for ¯
FUN (4) and ¯
FUN ( n ) , n ≥ Definition 21:
Consider the functional space ¯
FUN ( n ) ( ˜Π ( bicubic ) ), n ≥
4. For anedge with one boundary vertex, let n (cid:48) = (cid:26) if n = 4 n if n ≥ n (cid:48) −
3) ”side” middle controlpoints ˜ L , . . . , ˜ L n (cid:48) − , ˜ R , . . . , ˜ R n (cid:48) − and n (cid:48) − middle control points˜ C , . . . , ˜ C n (cid:48) − (see Figure 39(a), 39(b)).The ”Middle” system of equations consists of n (cid:48) equations: n (cid:48) − ”Eq(s)” for s = 2 , . . . , n (cid:48) and ”sumC-equation” (see Equation 63). The ”Restricted Middle” system of equations consists of n (cid:48) − ”Eq(s)” for s = 2 , . . . , n (cid:48) .Theorems 8 and 9 show that the ”central” middle control points are re-sponsible for the consistency of the ”Middle” system of equations and for thesatisfaction of ”sumC-equation” . Classification of the ”side” middle controlpoints is made after classification of the ”central” control points and dependson the rank of the ”Restricted Middle” system.The classification is based on the consistency and the rank analysis of the ”Middle” system. The relatively complicated algebraic proofs of the main resultsdeal with nice algebraic and geometric dependencies, which define the actual degrees of the weight functions and finally define the structure and the rank ofthe ”Middle” system. A necessary and sufficient condition for the consistency of the ”Middle” system. Classification of the ”central” middle control points.
Theorem 8:
Given a global in-plane parametrisation ˜Π ( bicubic ) ,for an edge withone boundary vertex, let all non-middle control points be classified and equations ”Eq(0)” , ”Eq(1)” , ”Eq(n’+1)” for n ≥ ”Eq(n’+2)” for n ≥ n ≥ ”C-equation” in a case when deg ( c ) − max deg ( l, r ) = 2 (70) (1) Is a sufficient condition for the satisfaction of the ”sumC-equation” . (2) Is a necessary and sufficient condition for the consistency of the ”Middle” system of equations.
Proof
See Appendix, Section D.
Lemma 23:
Consider the global in-plane parametrisation ˜Π ( bicubic ) . For an edgewith one boundary vertex, the ”central” middle control points are classifiedprior to and independently of the classification of the ”side” middle controlpoints according to the following rules. • ˜ C t for t = 3 , . . . , n (cid:48) − • ˜ C n (cid:48) − is a dependent control point with the dependency defined by ”C-equation” if deg ( c ) − max deg ( l, r ) = 2 and basic (free) otherwise. Note,that C n (cid:48) − may depend only on the basic ”central” middle control pointsAccording to Theorem 8, the classification guarantees that the ”sumC-equation” is satisfied and that the ”Middle” system of equations is consistent. Classification of the ”side” middle control points
Lemma 23 implies thatafter the classification of the ”central” middle control points, ”sumC-equation” issatisfied and it is sufficient to study the ”Restricted Middle” system of equations.The system is known to be consistent as a sub-system of the consistent ”Middle” system. The ”Restricted Middle” system contains 2( n (cid:48) −
3) non-classified ”side” middle control points ( ˜ L t , ˜ R t ) for t = 2 , . . . , n (cid:48) −
2. It remains to study the rankof the system and to classify the ”side” middle control points accordingly.The next Definition generalises the ”Projections Relation” (see Definition17) that plays an important role in the rank analysis of the ”Middle” system incase of global bilinear in-plane parametrisation ˜Π ( bilinear ) . Definition 22:
For an edge with one boundary vertex, let l ( v ), r ( v ) be con-ventional weight functions corresponding to a global in-plane parametrisation˜Π ( bicubic ) . Scalars g ( ij ) ( i, j ∈ { , , } ) are defined by the next formula in termsof coefficients of the weight functions g ( ij ) = l i r j − l j r i (71)The following three principal kinds of relations between g (01) , g (02) and g (12) will be considered. These relations correspond to the different possible values ofthe rank of the ”Restricted Middle” system of equations in terms of non-classified”side” middle control points. Theorem 9:
Given the global in-plane parametrisation ˜Π ( bicubic ) , for an edgewith one boundary vertex, let- All non-middle control points be classified and equations ”Eq(0)” , ”Eq(1)” , ”Eq(n’+1)” for n ≥ ”Eq(n’+2)” for n ≥ ”Restricted Middle” system in terms of the ”side” middlecontrol points and the classification of the ”side” middle control points dependon the relations between the g ( ij ) ”CASE 1” If g (01) g (12) g (02) (cid:54) = 0 and { g (02) } = 4 g (01) g (12) (72)then rank = n (cid:48) − n (cid:48) − n (cid:48) − n (cid:48) −
3) ”side” middle control points. ”CASE 2” If g (01) = g (12) = g (02) = 0 (73)then rank = n (cid:48) − n (cid:48) − n (cid:48) − L t , ˜ R t ) ( t = 2 , . . . , n (cid:48) −
2) onecontrol point is basic and one is dependent. ”CASE 0”
If none of previous conditions on g (01) , g (12) , g (02) are satisfied, then rank = n (cid:48) − n (cid:48) − n (cid:48) − n (cid:48) −
3) ”side” middle control points.Like in the case of global bilinear in-plane parametrisation ˜Π ( bilinear ) , explicitdependencies between the control points are described in the proof of the theo-rem. The different kinds of local templates
A local template for an edge withone boundary vertex includes the middle control points and is responsible forthe satisfaction of the ”Middle” system of equations. Application of a localtemplate assumes that all non-middle control points for the edge are classifiedand equations ”Eq(0)” , ”Eq(1)” , ”Eq(n’+1)” for n ≥ ”Eq(n’+2)” for n ≥ middle controlpoints is made as follows (see Figure 39(c)). ”Central” middle control points. These control points are classified prior toand independently of the classification of ”side” middle control points.According to Lemma 23, the local MDS contains all n (cid:48) − deg ( c ) − max deg ( l, r ) (cid:54) = 2 and contains n (cid:48) − ”Side” middle control points. Classification of these control points depends onthe rank of the ”Restricted Middle” system. The number of basic controlpoints varies from n (cid:48) − ”CASE 0” ) to n (cid:48) − ”CASE 2” ) (seeTheorem 9). Example of local MDS for n = 4 and n = 5 In both cases ( n = 4 or n = 5) alocal template contains the same control points: ˜ C , ˜ L , ˜ R , ˜ L , ˜ R .Lemma 23 implies that the control point ˜ C is dependent (and dependencyof C is defined by ”C-equation” ) if deg ( c ) − max deg ( l, r ) = 2 and belongs tolocal MDS otherwise.In ”CASE 0” all control points ˜ L , ˜ R , ˜ L , ˜ R are dependent. In ”CASE1” one of these four control points belongs to local MDS and others are de-pendent. In ”CASE 2” in every pair ( ˜ L , ˜ R ) and ( ˜ L , ˜ R ) one of the controlpoints belongs to local MDS and the second one is dependent. (a) (b) Fig. 38: Local templates for classification of V , E -type control points adjacentto a boundary vertex in the case of global in-plane parametrisation ˜Π ( bicubic ) . (a) (b) (c) Fig. 39: Local templates for an edge with one boundary vertex in the case ofglobal in-plane parametrisation ˜Π ( bicubic ) .
17 Global MDS17.1 Algorithm for construction
Algorithm 6 defines order of application of the local templates described inSection 16. The Algorithm allows to ”put together” local templates withoutcontradiction and to define global MDS. The Algorithm reuses Algorithms 1and 4 for the inner part of the mesh; local modifications for edges with oneboundary vertex are made at the last stage of the classification process.It was decided to present the common algorithm for construction of globalMDS of degree n = 4 and n ≥
5, because strategies for n = 4 and n ≥ Note 8:
MDS of degree 4 can be constructed in situations, when ”additional”constraints applied at a boundary vertex, involve only control points lying onthe global boundary (local template
T B ( bicubic ) , see Subsection 16.1.2, can beused for every boundary vertex). MDS of degree n ≥ Algorithm 6:
Algorithm for construction of global MDSin the case of global in-plane parametrisation ˜Π ( bicubic ) ”Stage 1” Classify all E , V , D , T -type control points adjacent to inner vertices exactly as it was done in the case of global bilinear in-plane parametrisa-tion ˜Π ( bilinear ) (see Subsections 11.1.3, 11.2.3, and Section 12).At the end of this stage, all ”Eq(0)”-type,”Eq(1)”-type equations for all inner vertices are satisfied.For an edge with one boundary vertex, control points ( ˜ L , ˜ C , ˜ R ), ( ˜ L , ˜ C , ˜ R )and ˜ C are classified and equations ”Eq(0)” and ”Eq(1)” are satisfied (seeFigure 39(a), 39(b)). ”Stage 2” For every edge with two inner vertices, classify the middle con-trol points exactly as it was done in the case of global bilinear in-planeparametrisation ˜Π ( bilinear ) (see Subsection 11.3)At the end of this stage all control points, lying on or adjacent to some edge with two inner vertices , and D -type control points adjacent to the inner vertex of an edge with one boundary vertex, are classified. All G -continuity equations excluding the ”Middle” system of equations andindexed equations ”Eq(n’+1)” for n ≥ ”Eq(n’+2)” for n ≥ ”Stage 3” For every edge with one inner vertex, perform the following twosteps of classification. ”Step 1”
At the boundary vertex apply local template (see Subsection16.1.2) - T B ( bicubic ) if a given ”additional” constraints, applied atthe vertex, involve only control points lying at the globalboundary and MDS of degree 4 is considered.-
T B ( bicubic ) if MDS of degree n ≥ middle control points are clas-sified and indexed equations ”Eq(0)”, ”Eq(1)”, ”Eq(n’+1)” for n ≥ ”Eq(n’+2)” for n ≥ ”Step 2” Apply local template for classification of the middle controlpoints.At the end of this step all control points are classified and all G -continuity equations are satisfied. Proof of the correctness
See Appendix, Section D. n ≥ Algorithm 6 allows to conclude that the following Lemma holds.
Lemma 24:
For any n ≥ B ( n ) ( ˜Π ( bicubic ) ) of the global MDS, that the instance fitsthe ”additional” constraints.Dimensionality of ˜ B ( n ) ( ˜Π ( bicubic ) ) is given in Subsection 17.4. n = 4 . Analysis ofdifferent ”additional” constraints. Lemma 25: (1)
Global ”pure”
MDS ˜ B (4) ( ˜Π ( bicubic ) ) (MDS which relates to G continuityconstraints alone) is well defined for any mesh configuration. (2) Let an ”additional” constraint, applied at any boundary vertex, involveonly the control points lying at the global boundary. Then an instance of˜ B (4) ( ˜Π ( bicubic ) ) , which fits the ”additional” constraint, exists for any meshconfiguration. In particular, a suitable ˜ B (4) ( ˜Π ( bicubic ) ) always exists fora (vertex)[boundary curve]-interpolation problem under the assumptionthat the (tangent plane)- interpolation is not required and for a partialdifferential equation with a simply-supported boundary condition.Dimensionality of ˜ B (4) ( ˜Π ( bicubic ) ) is presented in Subsection 17.4.There are two important differences with respect to the case of global bilin-ear in-plane parametrisation ˜Π ( bilinear ) . The first evident difference is that aninstance of global MDS of degree 4, which fits the (tangent plane)-interpolationcondition, is no longer known to exist. The second difference is that one nevertries to construct MDS of degree 4 in cases when its existence is not guaran-teed. In a case of the (tangent plane)-interpolation and a case of the clamped boundary condition either MDS of degree n ≥ ( bicubic ) , no D -relevant boundary vertices are considered (and therefore the main reason for fail-ure of MDS construction in the case of global bilinear parametrisation no longerexists), one encounters another problem. linearisation of the G -continuity con-dition for an edge with one boundary vertex leads to a more complicated ”Mid-dle” system of equations. Now construction of global MDS of degree 4 mayfail because of insufficient number of the control points which are necessary inorder to satisfy the ”Middle” system of equations for an edge with one boundaryvertex. The existence of a suitable MDS of degree 4 in the current case dependseven more strongly on the kind of ”additional” constraints than in the case ofglobal bilinear in-plane parametrisation. Theorem 10:
For global in-plane parametrisation ˜Π ( bicubic ) and for any n ≥ B ( n ) G (subset of MDS which participate in the G -continuity condi-tion) and dimension of ˜ B ( n ) (full dimension of MDS) are given by the followingformulas (see Definition 10 and Lemma 3) | ˜ B ( n ) G | = 3 | V ert non − corner | + (2 n − | Edge inner | + | V ert inner − regular | + | Edge two inner vertices, ” P rojections Relation ” holds |−| Edge one boundary vertex,deg ( c ) − max deg ( l, r ) = 2 | + | Edge one boundary vertex, ” CASE1 ” | + 2 | Edge one boundary vertex, ” CASE2 ” | (74) | ˜ B ( n ) | = | ˜ B ( n ) G | + | ˜ CP ( n ) F | = | ˜ B ( n ) G | + ( n − | F ace inner | + ( n − n − | F ace boundarynon − corner | +( n − | F ace corner | (75) Part VI. Mixed MDS of degrees and Definition of mixed MDS
For a global regular in-plane parametrisation ˜Π ∈ ˜ PAR ( m ) , m <
4, mixed functional space ¯
FUN (4 , ( ˜Π) is defined according toDefinition 4 where item (2) is substituted with the following item (2’) For a mesh element ˜ q ∈ ˜ Q , Z -coordinate of the restriction ¯ Q = ¯Ψ | ˜ q belongsto POL (4) if ˜ q is an inner mesh element and belongs to POL (5) if ˜ q is aboundary mesh element. Definition of MDS (see Definition ?? ) also requires just a minor modification. Definition 23: ˜ CP (4 , is composed of- Subset of ˜ CP (4) for inner mesh elements.- ”Degree 4” control points ˜ P (4) i,j , i = 0 , . . . , j = 0 , P (4)0 , , ˜ P (4)0 , and”degree 5” control points ˜ P (5) i, , i = 1 , . . . , P (5) i,j , i = 0 , . . . , j =3 , , Definition 24:
MDS ˜ B (4 , is such a minimal subset of ˜ CP (4 , that for any func-tion ¯Ψ ∈ ¯ FUN (4 , , equality to zero of Z -coordinates corresponding to all con-trol points from the subset implies that ¯Ψ ≡ Algorithm for the construction of mixed MDS ˜ B (4 , For both mesh with apolygonal global boundary (Part IV) and mesh with a smooth global boundary(Part V) a possible failure to construct MDS of degree 4, which fits a given”additional” constraints, is always related to the boundary mesh elements. Al-gorithm 7 (see Appendix, Section B) shows that in a case of failure it issufficient to elevate the degree up to 5 for the boundary patches only, leavingthe degree 4 for all inner patches. In other words, it is sufficient to pass over toconsideration of mixed MDS ˜ B (4 , .Clearly, Algorithm 7 results in lower than | ˜ B (5) | dimension of MDS. However,Algorithm 7 requires the global classification of D -type control points, likeAlgorithms for construction of MDS of degree 4. Although a formal proof of thecorrectness of the Algorithm is not given, one can easily verify it by her/himself. Algorithm 7:
See Appendix, Section B.
The existence of a suitable instance of mixed MDS for any type of ”addi-tional” constraints
Algorithm 7 allows to conclude that the following Lemmatakes place.
Lemma 26:
Let a planar mesh have either a polygonal or a smooth global bound-ary and global in-plane parametrisation ˜Π = ˜Π ( bilinear ) or ˜Π = ˜Π ( bicubic ) beconsidered. Then for any type of ”additional” constraints, there exists suchan instance of mixed MDS ˜ B (4 , ( ˜Π), that the instance fits the ”additional”constraints. Part VII. Computational examples
The current Part illustrates theoretical analysis given in the previous Parts by afew computational examples. The first example presents MDS for some irregular4-element mesh.The rest of the examples study more complicated irregular meshes; theseexamples present approximate solutions of the Thin Plate problem under dif-ferent boundary conditions. The main purpose of the examples is to verify theexistence of MDS that fits the boundary constraints and to demonstrate qualityof the resulting surface. Although some statistics connected to numerical pre-cision of the solution are given, an error estimate of the approximate solutionremains beyond the scope of the current research. )4(2 C~ )4(1 C~ )4(0 C~ )4(1 R )4(0 R )4(1 L~ )4(0 L~ )4(00 P~ )4(40 P~ )4(01 P~ )4(41 P~ )4(02 P~ )4(42 P~ (a) )5(0 C~ )5(2 C~ )5(1 C~ )5(1 R~ )5(0 R~ )5(1 L~ )5(0 L~ )5(03 P~ )5(04 P~ )5(05 P~ )5(53 P~ )5(54 P~ )5(55 P~ )5(12 P~ )5(42 P~ (b) Fig. 40: An illustration for degree elevation for a boundary mesh element.
18 Examples of MDS
Figure 41 presents an example of ”pure”
MDS ˜ B (4) for an irregular 4-elementmesh over a square domain. The basic control points are colored light blueand the dependent control points are colored red. The following control pointsbelong to ˜ B (4) . • ˜ CP (4) F contains 9 corner control points for every mesh element. • ˜ B (4) G contains the following control points- V -type control points. V -type control points for 5 non-corner vertices.- E -type control points. E -type control points for every one of the 5 non-corner vertices. (For non-corner boundary vertices, the inner and one ofthe boundary E -type control points belong to MDS).- D -type control points. D -type control points adjacent to the innervertex, the only D -relevant vertex for the considered mesh configuration.- T -type control points. One T -type control point for every one of 5 non-corner vertices.- Middle control points.
One ”side” middle control point for the left in-ner edge, since the ”Projections Relation” is satisfied for mesh elementsadjacent to the edge.According to Theorem 7, | ˜ B (4) G | = 3 × × | ˜ B (4) | = | ˜ B (4) G | + | ˜ CP (4) F | = 24 + 36 = 60 (76)Correctness of the construction of the MDS and of the formula given in Theorem7 was verified by the straightforward construction of such a linear system ofconstraints thatAll the control points of all the patches (100 control points altogether) arethe unknowns.The system of constraints is formed by G -continuity equations for allinner edges.The formal rank analysis of the system shows that its rank is equal to 40. Itmeans that there are 60 free unknowns, which is precisely equal to the numberof basic control points, obtained as a result of the construction of ”pure” MDS.The example shown in Figure 41(b) provides another illustration to thecorrectness of MDS construction. More precisely, it verifies the correctness ofthe classification rule for the middle control points (see Subsection 11.3). InFigure 41(b) the boundary vertex of the left inner edge is moved downwardswith respect to the example shown in Figure 41(a). Now the ”ProjectionsRelation” is no longer satisfied for the left edge and both middle control pointsadjacent to the edge are classified as dependent ones. It means that | ˜ B (4) | = 59,which again is precisely equal to the number of free unknowns resulting fromthe linear system of constraints for the current mesh configuration. (a) (b) Fig. 41: Examples of ˜ B (4) for irregular 4-element mesh.
19 Examples of an approximate solution of the Thin Plateproblem for irregular quadrilateral meshes
The example given on a circular domain can be compared to the recent survey[48] where several approches , including IGA with multi-patches are evaluated.One will note the complete different approach in the underlying mesh structure.
Let the middle plane of a plate with uniform thickness lie in ( XY )-plane. Thendeflection of the plate under load f - the solution of the Thin Plate problem -can be found by constrained minimisation of the energy functional E = (cid:82) (cid:82) ˜Ω (cid:20) D (cid:26)(cid:16) ∂ Z∂X + ∂ Z∂Y (cid:17) − − ν ) (cid:18) ∂ Z∂X ∂ Z∂Y − (cid:16) ∂ Z∂X∂Y (cid:17) (cid:19)(cid:27) − f Z (cid:21) dXdY (77)where the constraints are defined by simply-supported of clamped boundaryconditions. Here D = Eh / (cid:2) − ν ) (cid:3) (78)where h - thickness of the plate, ν - Poisson’s ratio and E - modulus of elasticity.References [46], [16], [7], [5] provide a detailed analysis of the ThinPlate problem and discuss different physical assumptions and limitations forapplication of Equation 77.Next two Subsections contain examples of approximate solutions of the ThinPlate problem, which are based on the technique presented in the current work.In both examples, a highly irregular meshing of a simple planar domain is con-sidered. It allows, on the one hand, to verify the correctness of the currentapproach for complicated meshes and, on the other hand, to compare the ap-proximate solution with the exact one, which is known for simple domains. In[48] , one will find an evaluation of many higher order methods on a similarproblem, noting that none are on a completely unstructured mesh. The precise deflection equation for a uniformly loaded circular thin plate withthe clamped boundary condition has the following form (see [51], [46]) Z ( X, Y ) = f ( R − r ) / (64 D ) (79)where R -radius of the circle, ˜ C = ( CX, CY ) - center of the circle, r = ( X − CX ) + ( Y − CY ) - square distance from a point to the center of the circle.An approximate solution of degree 5 is constructed for the irregular meshpresented in Figure 42(a). The mesh contains 20 quadrilateral elements; theglobal boundary of the domain is approximated by a piecewise-cubic parametricB´ezier curve (control points of the boundary curves are shown in Figure 42(a)).According to the current approach, global in-plane parametrisation ˜Π ( bicubic ) isconsidered. Control points of bicubic in-plane parametrisation for the boundarymesh elements are shown in Figure 42(b).In the example R = 1 in , ˜ C = (1 in, in ) and let h = 0 . in , ν = 0 . E = 40 × lb/in . From the formal point of view, f is just a coefficient of proportionality in the deflection equation. An unreasonably large value of f = 20 × lb/in is taken in order to visualize the resulting surface, while allother computations and statistics correspond to f = 20 lb/in .The resulting smooth surface is shown in Figure 43. The total number ofthe basic free control points is equal to 208. Level lines for the resulting surfaceand for its first-order derivatives (see Figure 46, Appendix, Section A) confirm C -smoothness of the surface. A comparison between the approximate andthe exact solutions along segment X = 1 in is shown in Figures 48 and 49(seeAppendix, Section A). One can see, that the approximate solution fits the exactone with a high precision. Errors in the center of the circle for the solution itselfand for the bending moments do not exceed one hundredth percent. Analysis of an approximate solution of degree 4 for a square domain with thesimply-supported boundary condition is made according to the same scenario.The exact solution for the uniform load is (see [51], [46]) Z ( X, Y ) = 16 f a π D ∞ (cid:88) m, n = 1 m, n − odd sin mπXa sin nπYb mn ( m + n ) (80)where the square domain is axes-aligned, the origin of the coordinate systemfor the XY -plane coincides with the lower-left corner of the square and a is thelength of the side.Let a = 2 in , h = 0 . in , ν = 0 . E = 40 × lb/in ; f = 5 × lb/in for visualization of the surface and f = 5 lb/in for comparison with the ex-act solution. An approximate solution is constructed for the irregular meshshown in Figure 44. In the current case global bilinear in-plane parametrisation˜Π ( bilinear ) is considered and it is sufficient to consider the space ¯ FUN (4) ( ˜Π ( bilinear ) ).The resulting smooth surface is shown in Figure 45. The total number ofthe basic free control points is equal to 111. Figure 47 (see Appendix, SectionA) presents the level lines for the surface and for its first-order derivatives.Comparison between the approximate and the exact solution along the segment X = 1 in is given in Figures 50 and 51 (see Appendix, Section A). It isimportant to note, that Figure 51 definitely shows that the approximate solutionis not C -smooth.Errors of the approximate solution for the irregular mesh were comparedwith the errors for the regular 4 × X = 1 in have the same order forthe meshes. Characteristic Maximal absolute valueof the error (in inches) Error in the center(in percents)Regularmesh Irregularmesh Regularmesh Irregularmesh Z . × − . × − . × − % 2 . × − % ∂Z∂X . × − . × − − − ∂Z∂Y . × − . × − − − M X . × − . × − .
26% 0 . M Y . × − . × − .
26% 0 . M XY . × − . × − − − (a) (b) Fig. 42: An irregular quadrilateral mesh for a circular domain.Fig. 43: The resulting smooth surface (case of the circular domain, irregularmesh). Fig. 44: Irregular quadrilateral mesh for a square domain.Fig. 45: The resulting smooth surface (case of the square domain, irregularmesh). Part VIII. Conclusions and further research.
A complete solution for cubic C boundaries has been constructed. Rigorouspoofs show that for n ≥ n = 4 has a solution provided we respect some conditions on the mesh.All the proofs are constructive and have been implemented for the examples wegave. We also have shown that one can mixes quartic and quintic patches. Asa conclusion from a practical point of view, the present work provides a way tosolve interpolation/approximation and partial differential problems for arbitrarystructures of quadrilateral meshes. The solution has a linear form, which makesit relatively simple, fast and stable. Construction of the MDS allows operat-ing only with essential degrees of freedom. Application of the approach to thesolution of fourth order partial differential equations results in an approximatesolution of a high quality.The following items are s a natural extension of the research, some trivial,others needing further studies: Boundary conditions
Boundary conditions ( BCs) on B´ezier or B-Splinesrequire special care, since the corresponding basis are non interpolator , cf. Embar et. all [1], and we have dealt only with the simple homogenouscase in our examples. The same remark apply to Robin type of BCs.Another topic,periodic BCs are not analysed here, but are essential intreating domains with holes.
Extension to non Rationnal B´ezier patches.
This seems simple for the caseof plane bilinear meshes , since the ”plane” part is trivially bilinear again,but the case of curved boundaries must be analysed anew.
Stability concerns.
Although the geometrical characteristics of a mesh playa very important role in the construction of MDS, they usually definethe general type of classification, while the choice of the basic controlpoints in possible cases of ambiguity is made arbitrarily. For example,for ˜ B (4) ( ˜Π ( bilinear ) , geometrical ”Projections Relation” defines the numberof the basic middle control points (see Section 11.3.2). However, thegeometry of the elements has no influence on the choice of the basic controlpoint when one of two middle control points is classified as basic andanother one as dependent. Taking into account mesh geometry in caseswhen an ambiguity is possible and introduction of the tolerance analysisinto equations themselves can significantly improve the stability of thesolution. Error estimation. . The results of Part VII demonstrate a high qualityof the solution. A rigorous mathematical analysis of the error in the caseof an approximate solution of a partial differential equation can be madeusing the error estimates methods of IGA, [26]
Combination of B´ezier patches with different polynomial degrees.
PartVI presents the construction of mixed MDS, which combines patches ofdegree 5 along the global boundary and patches of degree 4 in the inner part of the mesh. The approach may be improved so that fewer patches ofdegree 5 may be used. It should be possible to consider patches of degree5 locally, only in problematic areas of a mesh. Generalisation of MDS.
Higher order smoothness seems difficult to buildon unstructured quadrilateral meshes. But lower orders ( n ≤
4) shouldbe possible provide one uses macro elements A study of MDS for lowerpolynomial degrees of patches and/or higher orders of smoothness is avery important possible domain of the further research.
A study of non-planar initial meshes.
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Illustrations to approximate solution of the Thin Plate Problem A Illustrations to approximate solution of the Thin PlateProblem
Fig. 46: Level lines for the resulting surface and for its first-order derivatives (case of thecircular domain, irregular mesh) (a) Z (b) ∂Z∂X (c) ∂Z∂Y .Fig. 47: Level lines for the resulting surface and for its first-order derivatives (case of thesquare domain, irregular mesh) (a) Z (b) ∂Z∂X (c) ∂Z∂Y . Illustrations to approximate solution of the Thin Plate Problem Fig. 48: Values for approximate (bold line) and exact (thin line) solutions, their first orderderivatives and the bending moments along segment X = 1 (case of the circular domain,irregular mesh) (a) Z (b) ∂Z∂X (c) ∂Z∂Y (d) M X (e) M Y (f) M XY .Fig. 49: Difference between the approximate and the exact solutions (case of the circulardomain, irregular mesh) (a) Z (b) ∂Z∂X (c) ∂Z∂Y (d) M X (e) M Y (f) M XY . Illustrations to approximate solution of the Thin Plate Problem Fig. 50: Values for approximate (bold line) and exact (thin line) solutions, their first orderderivatives and the bending moments along segment X = 1 (case of the square domain,irregular mesh) (a) Z (b) ∂Z∂X (c) ∂Z∂Y (d) M X (e) M Y (f) M XY .Fig. 51: Difference between approximate and exact solutions (case of the square domain,irregular mesh) (a) Z (b) ∂Z∂X (c) ∂Z∂Y (d) M X (e) M Y (f) M XY . Algorithms B Algorithms
Algorithm 5
In order to clarify the Algorithm, the coefficient of a D -typecontrol point D in the ”Circular Constraint” corresponding to vertex ˜ V will bedenoted by coef f D ˜ V . Algorithm for construction of the D -dependency tree and classificationof D -type and T -type control points for a connected componentof the D -dependency graph which has no dangling half-edge Let connected component ˜ C of the D -dependency graph have no danglinghalf-edge. It means that ˜ C consists of D -relevant primary vertices and full edges only; let it contain | V | primary vertices and | E | ≥ | V | − • If | E | = | V | − then ˜ C coincides with its spanning tree and there is no chanceof assigning D -type control point to every vertex of the component (the Algo-rithm fails ). • Otherwise(1)
Build a spanning tree of ˜ C . There is at least one edge ˜ e of the connectedcomponent which does not participate in the spanning tree. (Although thechoice of the spanning tree and the choice of the edge may affect the stabilityand even the existence of the solution, the current presentation is restrictedto the description of a simple algorithm. A study of more complicated casesis left for a possible further research). (2) Choose one vertex of ˜ e to be the root of the D -dependency tree (vertex ˜ R ),denote the second vertex by ˜ V (0) (see Figure 52). Assign the D -type controlpoint of edge ˜ e to the root vertex ˜ R (denote the control point by ˜ D ( ˜ R )). (3) Build the D -dependency tree of ˜ C : direct every mesh edge of the tree fromthe upper vertex to a lower one (according to the hierarchy of the spanningtree). (4) Classify D -type and T -type control points according to the same principlesas in Algorithm 4. The only difference is that now ˜ D ( ˜ R ) can not be con-sidered as a basic control point. Let ˜ V (0) → ˜ V (1) → . . . ˜ V ( k ) → ˜ R be thepath in the D -dependency tree from vertex ˜ V (0) to the root. All vertices˜ V (0) , . . . , ˜ V ( k ) are inner even vertices of the initial mesh, because any bound-ary vertex uses at most one D -type control point and so it has at most oneadjacent full edge in the D -dependency tree. The following local modifica-tions of Algorithm 4 should be made. (a) According to Algorithm 4, one relates to ˜ D ( ˜ R ) for the first time whenthe level of vertex ˜ V (0) in the D -dependency tree (which is traverseddown-up) is reached. At this step one should ”close” the ”CircularConstraint” of vertex ˜ V (0) using D ( ˜ V (0) ). Both D ( ˜ R ) and D ( ˜ V (0) )participate in the ”Circular Constraint” with a non-zero coefficients, be-cause both corresponding full edges belong to the D -dependency graph.Therefore the ”Circular Constraint” defines dependency of D ( ˜ V (0) ) on D ( ˜ R ) in the following way Algorithms D ( ˜ V (0) ) = − coeff D ( ˜ R )˜ V (0) coeff D ( ˜ V (0))˜ V (0) D ( ˜ R ) + f D ( ˜ V (0) ) (81)where f D ( ˜ V (0) ) stands for some linear combination of the basic controlpoints. (b) Proceeding to traverse the D -dependency tree down-up, one graduallydefines the linear dependencies of D ( ˜ V (1) ) , . . . , D ( ˜ V ( k ) ) on D ( ˜ R ) andsome basic control points. At the level of the root vertex one gets D ( ˜ R ) = coef f D ( ˜ R ) + f D ( ˜ R ) (82)where coef f = ( − k coeff D ( ˜ R )˜ V (0) coeff D ( ˜ V (0))˜ V (0) coeff D ( ˜ V (0))˜ V (1) coeff D ( ˜ V (1))˜ V (1) . . . coeff D ( ˜ V ( k − V ( k ) coeff D ( ˜ V ( k ))˜ V ( k ) (83) (c) If coef f (cid:54) = 1 (which holds in a general case) then ˜ D ( ˜ R ) is classified asa dependent control point; its dependency on the basic control points isdefined according to Equation 82. It remains to substitute the expres-sion for D ( ˜ R ) into dependency equations for D ( ˜ V (0) ) , . . . , D ( ˜ V ( k ) ). Inthis case, the Algorithm succeeds .Otherwise it is possible either to try to choose another ”free” edge of thespanning tree or even another spanning tree and to run the Algorithmfrom the beginning or to decide that the Algorithm fails and to use oneof the possibilities listed in the end of Subsection 12.2.3. Algorithm 7
Algorithm for construction of mixed MDS ˜ B (4 , Let a planar mesh have either a polygonal or a smooth global boundary andglobal in-plane parametrisation ˜Π = ˜Π ( bilinear ) or ˜Π = ˜Π ( bicubic ) is considered.In description of the Algorithm superscripts (4) and (5) are added in order tospecify whether some equation or control point relates to degree 4 or 5 of theresulting patch. ”Stage 1” Assume (artificially) that there are no D -relevant boundary ver-tices. Classify all V (4) , E (4) , D (4) , T (4) -type control points adjacent to the inner vertices precisely as in the case of global bilinear in-plane parametrisation˜Π ( bilinear ) (see Subsections 11.1.3 and Algorithm 4). The classification existsaccording to Theorem 6.At the end of this stage all V (4) , E (4) , T (4) -type control points for all innervertices and all D (4) -type control points are classified and ”Eq(0)” -type and ”Eq(1)” -type equations at all inner vertices are satisfied.For an edge with one boundary vertex, control points ( ˜ L (4)0 , ˜ C (4)0 , ˜ R (4)0 ),( ˜ L (4)1 , ˜ C (4)1 , ˜ R (4)1 ) and ˜ C (4)2 are classified and equations ” Eq (4) (0)” and ” Eq (4) (1)”are satisfied (see Figure 40(a)). ”Stage 2” For every inner edge with two inner vertices, apply a local templatefor classification of the middle (4) control points, exactly as in the case of globalbilinear in-plane parametrisation ˜Π ( bilinear ) (see Subsection 11.3). Algorithms At the end of this stage all points from ˜ CP (4) G lying on or adjacent to any edge with two inner vertices and all D (4) -type control points are classified (seeFigure 40(a)). G -continuity conditions are satisfied for any edge with twoinner vertices. For every boundary mesh element, all ”degree 4” control pointsare classified. ”Stage 3” For every boundary mesh element, consider 3 D patch of degree5. Use the degree elevation formulas in order to classify control points ˜ P (5) i,j , i = 0 , . . . , j = 0 , P (5)0 , , ˜ P (5)5 , and in order re-compute their dependencieson ”degree 4” basic control points from ˜ CP (4 , (see Figure 40).At the end of this stage, all G -continuity conditions hold for all edgeswith two inner vertices . For an edge with one boundary vertex, control points( ˜ L (5) i , ˜ C (5) i , ˜ R (5) i ), i = 0 , C (5)2 are classified and equations ” Eq (5) (0)” and” Eq (5) (1)” are satisfied (see Technical Lemma 7, Appendix, Section C) ”Stage 4” For an edge with one boundary vertex, locally classify control points( ˜ L (5) i , ˜ R (5) i ) ( i = 2 , , ,
5) and ˜ C (5) i ( i = 3 , ,
5) in such a manner that theremaining G -continuity equations for the edge will be satisfied.In the case of a mesh with a polygonal global boundary and global in-planeparametrisation ˜Π ( bilinear ) - V , E -type control points adjacent to the boundary vertex ( ˜ L (5)5 , ˜ C (5)5 , ˜ R (5)5 and C (5)4 ) are classified by a local template described in Subsection 11.1.4.- D , T -type control points adjacent to the boundary vertex ( L (5)4 , R (5)4 and C (5)3 ) are classified by a local template described in Subsection 11.2.4.- Middle control points ( ˜ L (5)2 , ˜ R (5)2 , ˜ L (5)3 , ˜ R (5)3 ) are classified according to alocal template described in Subsection 11.3.In the case of a mesh with a smooth global boundary and global in-planeparametrisation ˜Π ( bicubic ) - Control points ˜ L (5)5 , ˜ C (5)5 , ˜ R (5)5 , i = 4 , T B ( bicubic ) , described in Subsection 16.1.2.- Middle control points ˜ L (5)3 , ˜ C (5)3 , ˜ R (5)3 , ˜ L (5)2 , ˜ R (5)2 are classified accordingto a local template described in Subsection 16.2.2. Algorithms Fig. 52: An illustration for the construction of a D -dependency tree when acomponent of D -dependency graph has no dangling half-edge. Technical Lemmas C Technical Lemmas
Technical Lemma 1:
Let ˜Π ( bilinear ) ∈ ˜ PAR (1) be global bilinear parametrisationand let ˜ λ ,˜ λ (cid:48) ,˜ γ ,˜ γ (cid:48) ,˜ ρ ,˜ ρ (cid:48) be vertices of two adjacent mesh elements (see Figure12). Then control points, which belong to ˜ CP ( n ) G ( ˜Π ( bilinear ) ) and participate in G -continuity conditions for the common edge of two patches, and first-orderdifferences of the control points have the following representations˜ C j = ˜ γ (cid:0) − jn (cid:1) + jn ˜ γ (cid:48) , ˜ R j = n − n (cid:0) ˜ γ n − jn + ˜ γ (cid:48) jn (cid:1) + n (cid:0) ˜ ρ n − jn + ˜ ρ (cid:48) jn (cid:1) , ˜ L j = n − n (cid:0) ˜ γ n − jn + ˜ γ (cid:48) jn (cid:1) + n (cid:16) ˜ λ n − jn + ˜ λ (cid:48) jn (cid:17) , ∆ ˜ C j = n (˜ γ (cid:48) − ˜ γ ) , ∆ ˜ L j = − n (cid:16) (˜ λ − ˜ γ ) n − jn + (˜ λ (cid:48) − ˜ γ (cid:48) ) jn (cid:17) , ∆ ˜ R j = − n (cid:0) (˜ ρ − ˜ γ ) n − jn + (˜ ρ (cid:48) − ˜ γ (cid:48) ) jn (cid:1) (84)Here j = 0 , . . . , n in expressions for ˜ C j , ˜ R j , ˜ L j , ∆ L j , ∆ R j and j = 0 , . . . , n − C j . Technical Lemma 2:
The control points of bicubic in-plane parametrisation ˜ P ( u, v ) = (cid:80) i,j =0 ˜ P ij B ij ( u, v ) for a boundary mesh element in the case of a planar meshwith a piecewise-cubic G -smooth global boundary have the following explicitformulas in terms of the planar mesh data (see Figure 34)˜ P = ˜ A ˜ P = ( ˜ A + 2 ˜ D )˜ P = (2 ˜ A + ˜ B ) ˜ P = (2 ˜ A + ˜ B + ˜ C + 2 ˜ D + 3 ˜ E )˜ P = ( ˜ A + 2 ˜ B ) ˜ P = ( ˜ A + 2 ˜ B + 2 ˜ C + ˜ D + 3 ˜ F )˜ P = ˜ B ˜ P = ( ˜ B + 2 ˜ C )˜ P = (2 ˜ A + ˜ D ) ˜ P = ˜ D ˜ P = (4 ˜ A + 2 ˜ B + ˜ C + 2 ˜ D ) ˜ P = ˜ E ˜ P = (2 ˜ A + 4 ˜ B + 2 ˜ C + ˜ D ) ˜ P = ˜ F ˜ P = (2 ˜ B + ˜ C ) ˜ P = ˜ C (85) Technical Lemma 3:
Let ˜ P ( u, v ) be the bicubic in-plane parametrisation of aboundary mesh element defined according to Technical Lemma 2 (see Figure34). Let˜ e = (cid:110) (2 ˜ A + ˜ B ) + ( ˜ C + 2 ˜ D ) (cid:111) ˜ f = (cid:110) ( ˜ A + 2 ˜ B ) + (2 ˜ C + ˜ D ) (cid:111) (86)and let two families of the planar vectors ˜ g ( u ) i and ˜ g ( v ) j ( i, j = 0 , . . . ,
3) be defined
Technical Lemmas as follows (see Figure 53(a))˜ g ( u )0 = ˜ B − ˜ A ˜ g ( v )0 = ˜ D − ˜ A ˜ g ( u )1 = ˜ E − ˜ D ˜ g ( v )1 = ˜ E − ˜ e ˜ g ( u )2 = ˜ F − ˜ E ˜ g ( v )2 = ˜ F − ˜ f ˜ g ( u )3 = ˜ C − ˜ F ˜ g ( v )3 = ˜ C − ˜ B (87)Let further ˜ G ( u ) and ˜ G ( v ) be the minimal infinite sectors which respectivelycontain all vectors from the first and second families, and let these sectors bebounded by rays corresponding to the angles α ( u ) min , α ( u ) max , α ( v ) min , α ( v ) max (seeFigure 53(b)). Then (1) (cid:110) ˜ g ( u ) i (cid:111) i =0 and (cid:110) ˜ g ( u ) j (cid:111) j =0 are respectively generators of ∂ ˜ P∂u and ∂ ˜ P∂v inthe meaning that for every parametric value ( u, v ) ∈ [0 , , ∂ ˜ P∂u and ∂ ˜ P∂v can be written as a positive linear combination of these vectors. Here apositive linear combination is defined as such a linear combination thatall its coefficients are non-negative and at least one coefficient is strictlypositive. (2)
Relations α ( u ) min > − α ( v ) max and α ( u ) max < α ( v ) min provide a sufficient conditionfor the regularity of parametrisation ˜ P ( u, v ). Proof(1)
The first-order partial derivatives of in-plane parametrisation have the fol-lowing representations in terms of the control points ˜ P ij , ( i = 0 , . . . , , j =0 , . . . , ∂ ˜ P∂u ( u, v ) = 3 (cid:80) i =0 (cid:80) j =0 ( ˜ P i +1 ,j − ˜ P ij ) B i ( u ) B j ( v ) ∂ ˜ P∂v ( u, v ) = 3 (cid:80) i =0 (cid:80) j =0 ( ˜ P i,j +1 − ˜ P ij ) B i ( u ) B j ( v ) (88)B´ezier polynomials are always non-negative and for every parametric value( u, v ) ∈ [0 , ], at least one of the products (cid:8) B i ( u ) B j ( v ) (cid:9) i = 0 , . . . , j = 0 , . . . , is strictlypositive. It implies that in order to prove that (cid:110) g ( u ) i (cid:111) i =0 and (cid:110) g ( v ) j (cid:111) j =0 serveas generators of ∂ ˜ P∂u and ∂ ˜ P∂v , it is sufficient to show that every one of the first-order differences of the control points can be represented as a positive linearcombination of the vectors from the corresponding family. Explicit formulas forthe control points of in-plane parametrisation (see Equation 85) lead to thefollowing expressions.
Technical Lemmas The first-order differences in u -direction˜ P − ˜ P = ˜ P − ˜ P = ˜ P − ˜ P = ( ˜ B − ˜ A ) = ˜ g ( u )0 ˜ P − ˜ P = ˜ P − ˜ P = ˜ P − ˜ P = (2( ˜ B − ˜ A ) + ( ˜ C − ˜ D )) = (2˜ g ( u )0 + ˜ g ( u )1 + ˜ g ( u )2 + ˜ g ( u )3 )˜ P − ˜ P = (( ˜ B − ˜ A ) + ( ˜ C − ˜ D ) + 3( ˜ E − ˜ D )) = (˜ g ( u )0 + 4˜ g ( u )1 + ˜ g ( u )2 + ˜ g ( u )3 )˜ P − ˜ P = (( ˜ B − ˜ A ) + ( ˜ C − ˜ D ) + 3( ˜ F − ˜ E )) = (˜ g ( u )0 + ˜ g ( u )1 + 4˜ g ( u )2 + ˜ g ( u )3 )˜ P − ˜ P = (( ˜ B − ˜ A ) + ( ˜ C − ˜ D ) + 3( ˜ C − ˜ F )) = (˜ g ( u )0 + ˜ g ( u )1 + ˜ g ( u )2 + 4˜ g ( u )3 )˜ P − ˜ P = ˜ E − ˜ D = ˜ g ( u )1 ˜ P − ˜ P = ˜ F − ˜ E = ˜ g ( u )2 ˜ P − ˜ P = ˜ C − ˜ F = ˜ g ( u )3 (89)The first-order differences in v -direction˜ P − ˜ P = ˜ P − ˜ P = ˜ P − ˜ P = ( ˜ D − ˜ A ) = ˜ g ( v )0 ˜ P − ˜ P = (2( ˜ D − ˜ A ) + ( ˜ C − ˜ B )) = (2˜ g ( v )0 + ˜ g ( v )1 )˜ P − ˜ P = (6( ˜ E − ˜ e ) + 2( ˜ D − ˜ A ) + ( ˜ C − ˜ B ) = (2 g ( v )0 + 6 g ( v )1 + g ( v )3 )˜ P − ˜ P = ( ˜ E − ˜ e ) = ˜ g ( v )1 ˜ P − ˜ P = (( ˜ D − ˜ A ) + 2( ˜ C − ˜ B )) = (˜ g ( v )0 + 2˜ g ( v )1 )˜ P − ˜ P = (6( ˜ F − ˜ f ) + ( ˜ D − ˜ A ) + 2( ˜ C − ˜ B ) = ( g ( v )0 + 6 g ( v )2 + 2 g ( v )3 )˜ P − ˜ P = ( ˜ F − ˜ f ) = ˜ g ( v )2 ˜ P − ˜ P = ˜ P − ˜ P = ˜ P − ˜ P = ( ˜ C − ˜ B ) = ˜ g ( v )3 (90)Relations 89 and 90 imply that for every parametric value of ( u, v ) ∈ [0 , ,the first-order partial derivatives ∂ ˜ P∂u ( u, v ) and ∂ ˜ P∂v ( u, v ) can be represented as apositive linear combination of the generators. It completes the straightforwardproof of the first statement of Technical Lemma 3. (2) The proof of the regularity of the bicubic in-plane parametrisation is sub-divided into a few parts according to the different requirements listed in thedefinition of a regular parametrisation (Definition 8).
Bijection.
One should prove that ˜ P ( u, v ) defines a bijection. On the con-trary, let there exist such parametric values ( u , v ) (cid:54) = ( u , v ) that ˜ P ( u , v ) =˜ P ( u , v ). Let segment σ ( t ) connect these two values in the parametric domain σ ( t ) = ( u , v )(1 − t ) + ( u , v ) t (91)Then ˜ P ( t ) = ˜ P ( σ ( t )) is a smooth closed curve in XY -plane. Let f ( t ) be Z -coordinate of the vector product of ˜ P ( t ) and some constant vector ˜ df ( t ) = (cid:104) ˜ d, ˜ P ( t ) (cid:105) (92) Technical Lemmas The choice of ( u , v ) and ( u , v ) implies that f (0) = f (1). According to theLagrange theorem, there exists such a value τ ∈ [0 ,
1] that f (cid:48) ( τ ) = 0.In order to show that the assumption is not correct, it is sufficient to choosesuch a vector ˜ d that f (cid:48) ( t ) (cid:54) = 0 for every t ∈ [0 , d is based on the formula f (cid:48) ( t ) = (cid:104) ˜ d, ∂ ˜ P∂u ( t )( u − u ) + ∂ ˜ P∂v ( t )( v − v ) (cid:105) (93)and is made in the following manner- If ( u − u )( v − v ) ≥ d strictly between − α ( v ) max and α ( u ) min . In this case (cid:104) ˜ d, ∂ ˜ P∂u (cid:105) > (cid:104) ˜ d, ∂ ˜ P∂v (cid:105) > t ; f (cid:48) ( t )for every t has the same sign as ( u − u ) (or as ( v − v ) if ( u − u ) = 0)and can not be equal to zero.- If ( u − u )( v − v ) < d strictly between α ( u ) max and α ( v ) min . In this case (cid:104) ˜ d, ∂ ˜ P∂u (cid:105) < (cid:104) ˜ d, ∂ ˜ P∂v (cid:105) > t and f (cid:48) ( t )for every t has the same sign as ( v − v ). Smoothness.
It is clear that ˜ P ( u, v ) is C -smooth as a bicubic B´ezier polyno-mial. Regularity of Jacobian.
Note, that det ( J ( ˜ P ) ( u, v )) = (cid:104) ∂ ˜ P∂u , ∂ ˜ P∂v (cid:105) . The regularityof the Jacobian J ( ˜ P ) trivially follows from the facts that for every parametricvalue ( u, v ) ∈ [0 , , the partial derivatives ∂ ˜ P∂u and ∂ ˜ P∂v can be representedas some positive linear combinations of generators (cid:110) ˜ g ( u ) i (cid:111) i =0 and (cid:110) ˜ g ( v ) j (cid:111) j =0 respectively and that (cid:104) ˜ g ( u ) i , ˜ g ( v ) j (cid:105) > i, j = 0 , . . . , G ( u ) and ˜ G ( v ) . (cid:116)(cid:117) TechnicalLemma 3
Technical Lemma 4:
Let a planar mesh have piecewise-cubic G -smooth globalboundary and ˜ P ( u, v ) be the restriction of the global in-plane parametrisation˜Π ( bicubic ) on such a boundary mesh element, that its upper edge lies on theglobal boundary (see Figure 34). Then the-first order partial derivatives of˜ P ( u, v ) along the left and the right edges of the element have the followingexplicit form.The partial derivative in the direction along the left edge ∂ ˜ P∂v (0 , v ) = ˜ D − ˜ A (94)The partial derivative in the cross direction for the left edge ∂ ˜ P∂u (0 , v ) = ( ˜ B − ˜ A ) B ( v ) + (cid:16) ( ˜ B − ˜ A )+( ˜ C − ˜ D ) (cid:17) B ( v ) + 3( ˜ E − ˜ D ) B ( v ) (95)The partial derivative in the direction along the right edge ∂ ˜ P∂v (1 , v ) = ˜ C − ˜ B (96) Technical Lemmas The partial derivative in the cross direction for the right edge ∂ ˜ P∂u (1 , v ) = ( ˜ B − ˜ A ) B ( v ) + (cid:16) ( ˜ B − ˜ A )+( ˜ C − ˜ D ) (cid:17) B ( v ) + 3( ˜ C − ˜ F ) B ( v ) (97) Technical Lemma 5:
Let conventional weight functions c ( v ), l ( v ), r ( v ) for thecommon edge of two adjacent boundary mesh elements be defined according toglobal in-plane parametrisation ˜Π ( bicubic ) . Then the coefficients of the weightfunctions have the following representations in terms of the initial mesh data(see Figure 35) l = (cid:104) ˜ ρ − ˜ γ, ˜ γ (cid:48) − ˜ γ (cid:105) l = (cid:104) (˜ ρ − ˜ γ ) + (˜ ρ (cid:48) − ˜ γ (cid:48) ) , ˜ γ (cid:48) − ˜ γ (cid:105) l = 3 (cid:104) ˜ T (cid:48) ρ − ˜ γ (cid:48) , ˜ γ (cid:48) − ˜ γ (cid:105) r = −(cid:104) ˜ γ − ˜ λ, ˜ γ (cid:48) − ˜ γ (cid:105) r = − (cid:104) (˜ γ − ˜ λ ) + (˜ γ (cid:48) − ˜ λ (cid:48) ) , ˜ γ (cid:48) − ˜ γ (cid:105) r = − (cid:104) ˜ γ (cid:48) − ˜ T (cid:48) λ , ˜ γ (cid:48) − ˜ γ (cid:105) (98) c = (cid:104) ˜ γ − ˜ λ, ˜ ρ − ˜ γ (cid:105) c = (cid:16) (cid:104) ˜ γ − ˜ λ, ˜ ρ (cid:48) − ˜ γ (cid:48) (cid:105) + (cid:104) ˜ γ (cid:48) − ˜ λ (cid:48) , ˜ ρ − ˜ γ (cid:105) +2 (cid:104) ˜ γ − ˜ λ, ˜ ρ − ˜ γ (cid:105) (cid:17) c = (cid:16) (cid:104) ˜ γ − ˜ λ, ˜ T (cid:48) ρ − ˜ γ (cid:48) (cid:105) + 3 (cid:104) ˜ γ (cid:48) − ˜ T (cid:48) λ , ˜ ρ − ˜ γ (cid:105) + (cid:104) (˜ γ − ˜ λ ) + (˜ γ (cid:48) − ˜ λ (cid:48) ) , (˜ ρ − ˜ γ ) + (˜ ρ (cid:48) − ˜ γ (cid:48) ) (cid:105) (cid:17) c = (cid:16) (cid:104) (˜ γ − ˜ λ ) + (˜ γ (cid:48) − ˜ λ (cid:48) ) , ˜ T (cid:48) ρ − ˜ γ (cid:48) (cid:105) + (cid:104) ˜ γ (cid:48) − ˜ T (cid:48) λ , (˜ ρ − ˜ γ ) + (˜ ρ (cid:48) − ˜ γ (cid:48) ) (cid:105) (cid:17) c = 9 (cid:104) ˜ γ (cid:48) − ˜ T (cid:48) λ , ˜ T (cid:48) ρ − ˜ γ (cid:48) (cid:105) (99) Technical Lemma 6:
Let conventional weight functions c ( v ), l ( v ), r ( v ) for thecommon edge of two adjacent boundary mesh elements be defined according toglobal in-plane parametrisation ˜Π ( bicubic ) . Then the following relations betweenthe coefficients of the weight functions with respect to B´ezier and the powerbases hold l ( power )0 = l r ( power )0 = r l ( power )1 = 2( l − l ) r ( power )1 = 2( r − r ) l ( power )2 = l − l + l r ( power )2 = r − r + r (100) c ( power )0 = c c ( power )1 = 4( − c + c ) c ( power )2 = 6( c − c + c ) c ( power )3 = 4( − c + 3 c − c + c ) c ( power )4 = c − c + 6 c − c + c (101) Technical Lemmas Technical Lemma 7:
Let a planar mesh have either a polygonal or a smoothglobal boundary and let global in-plane parametrisation ˜Π ( bilinear ) or ˜Π ( bicubic ) be considered. Further, for the inner vertex of an edge with one boundary vertex,let the system of all ” Eq (4) (0)” -type and ” Eq (4) (1)” -type equations be satisfiedby classification of V (4) -type, E (4) -type, D (4) -type and T (4) -type control pointsadjacent to the vertex. In particular, for the edge with one boundary vertex,control points ˜( L (4) i , ˜ C (4) i , ˜ R (4) i ) for i = 0 , C (4)2 are classified (see Figure40(a)). Then (1) The degree elevation formulas for control points ( ¯ L (5) i , ¯ C (5) i , ¯ R (5) i ) for i =0 , C (5)2 (see Figure 40(b)) involve only control points ( ¯ L (4) i , ¯ C (4) i , ¯ R (4) i )for i = 0 , C (4)2 . (2) Control points ( L (5) i , C (5) i , R (5) i ) ( i = 0 ,
1) and C (5)2 , computed according tothe degree elevation formulas, satisfy equations ” Eq (5) (0)” and ” Eq (5) (1)”for the edge with one boundary vertex.Here superscripts (4) and (5) are used in order to specify whether some equationor control point relates to degree 4 or 5 of the resulting patch. Proof(1)
The first statement of the Technical Lemma immediately follows from thestraightforward formulas¯ R (5)0 = 1 / R (4)0 + ¯ C (4)0 )¯ C (5)0 = ¯ C (4)0 ¯ L (5)0 = 1 / L (4)0 + ¯ C (4)0 )¯ R (5)1 = 1 / R (4)1 + 4 ¯ R (4)0 + 4 ¯ C (4)1 + ¯ C (4)0 )¯ C (5)1 = 1 / C (4)1 + ¯ C (4)0 )¯ L (5)1 = 1 / L (4)1 + 4 ¯ L (4)0 + 4 ¯ C (4)1 + ¯ C (4)0 )¯ C (5)2 = 1 / C (4)2 + 2 ¯ C (4)1 ) (102)The important implication is that as soon as the control points ( ˜ L (4) i , ˜ C (4) i , ˜ R (4) i )( i = 0 ,
1) and ˜ C (4)2 are classified, control points ( ˜ L (5) i , ˜ C (5) i , ˜ R (5) i ) ( i = 0 ,
1) and˜ C (5)2 are also classified and dependencies of the dependent control points from˜ CP (5) G are fully defined. (2) It is sufficient to prove the second statement of the Technical Lemma inthe case when the conventional weight functions correspond to global bilinearin-plane parametrisation ˜Π ( bicubic ) . Indeed, Lemma 21 shows that for an edgewith one boundary vertex, the system of equations ”Eq(0)” and ”Eq(1)” in thecase of ˜Π ( bicubic ) is equivalent to the system in the case of ˜Π ( bilinear ) .Equations ” Eq (5) (0)” and ” Eq (5) (1)” involve coefficients of the weight func-tions (which clearly do not depend on chosen degree 4 or 5 of MDS) and the first-order differences of the control points (∆ L (5) i , ∆ C (5) i , ∆ R (5) i ), i = 0 , L (4) i , ∆ C (4) i , ∆ R (4) i ), i = 0 , Technical Lemmas Eq (5) (0)” = 4 / Eq (4) (0)”” Eq (5) (0)” = 4 / (cid:0) ” Eq (4) (0)” + ” Eq (4) (1)” (cid:1) (103)The last two equations clearly imply that statement (2) of the Technical Lemmais satisfied. (cid:116)(cid:117) Technical Lemma 7 (a) (b)
Fig. 53: An illustration for a sufficient condition for the regularity of bicubicin-plane parametrisation.
Proofs D Proofs
Proof of Theorem 3
First, it will be shown that the satisfaction of Equation 17 with scalarfunctions given in Equation 18 is sufficient in order to guarantee a G -smoothconcatenation between the adjacent patches. It will be verified, that the scalarfunctions given in Equation 18 satisfies all requirements of Definition 7 andmay be used as the weight function. Equation 6.
It should be verified that Equation 6 is satisfied for X and Y components of the partial derivatives ¯ L u , ¯ R u and ¯ L v . For example, for X -component, one gets LX u ( v ) l ( v ) + RX u ( v ) r ( v ) + LX v ( v ) c ( v ) = LX u < ˜ R u , ˜ L v > − RX u < ˜ L u , ˜ L v > + LX v < ˜ L u , ˜ R u > = mix ˜ L u LX u ˜ R u RX u ˜ L v LX v = det LX u LY u LX u RX u RY u RX u LX v LY v LX v = 0 . (104)The same proof works for Y -components of the partial derivatives. Equation 7.
It is necessary to verify that l ( v ) r ( v ) < v ∈ [0 , l ( v ) r ( v ) = 0 is impossible as a trivial consequence of the regularity ofparametrisation. Indeed, for example equality to zero of r ( v ) implies that0 = r ( v ) = − < ˜ L u , ˜ L v > = − det (cid:18) ∂LX∂u ∂LY∂u∂LX∂v ∂LY∂v (cid:19) = − det ( J (˜ L ) (1 , v )) (105)that contradicts the regularity of parametrisation for the left patch. l ( v ) r ( v ) is a continuous function and l (0) = < ˜ R u (0) , ˜ L v (0) >> r (0) = − < ˜ L u (0) , ˜ L v (0) >< R u (0) and ˜ L u (0) clearly point to theright side with respect to the tangent vector of the common boundary ˜ L v (0).Therefore l ( v ) r ( v ) ≤ v ∈ [0 , Equation 8.
One should show that < ¯ L u , ¯ L v > (cid:54) = 0 for every v ∈ [0 , Z -component of this cross product isequal to < ˜ L u , ˜ L v > = − r ( v ) which is already proven to be non-zero.Now it will be shown that if the adjacent patches join the G -smoothly, thenEquation 17 with scalar functions given in Equation 18 is necessarily satisfied.Let the adjacent patches join the G -smoothly. Then, according to Definition7, there exist such scalar functions ˘ l ( v ), ˘ r ( v ), ˘ c ( v ) that Equations 6, 7, 8 aresatisfied. In particular, Equation 6 implies that XY -components of the partialderivatives satisfy the equation˜ L u ˘ l ( v ) + ˜ R u ˘ r ( v ) + ˜ L v ˘ c ( v ) = 0 (106)The result of the cross product of the last expression respectively with ˜ L v and˜ R u is clearly equal to zero. Together with the inequality (cid:104) ˜ R u , ˜ L v (cid:105) (cid:54) = 0 (cid:105) (which Proofs follows from the regularity of in-plane parametrisation), it leads to the conclu-sion that the following relations between ˘ l ( v ), ˘ r ( v ), ˘ c ( v ) necessarily hold˘ r ( v ) = − (cid:104) ˜ L u , ˜ L v (cid:105)(cid:104) ˜ R u , ˜ L v (cid:105) ˘ l ( v ) , ˘ c ( v ) = (cid:104) ˜ L u , ˜ R u (cid:105)(cid:104) ˜ R u , ˜ L v (cid:105) ˘ l ( v ) (107)and Equation 6 necessarily has the form¯ L u ˘ l ( v ) − ¯ R u (cid:104) ˜ L u , ˜ L v (cid:105)(cid:104) ˜ R u , ˜ L v (cid:105) ˘ l ( v ) + ¯ L v (cid:104) ˜ L u , ˜ R u (cid:105)(cid:104) ˜ R u , ˜ L v (cid:105) ˘ l ( v ) = 0 (108)According to Equation 7, ˘ l ( v ) (cid:54) = 0 for any v ∈ [0 , ˘ l ( v ) (cid:104) ˜ R u , ˜ L v (cid:105) . Therefore one sees, thatEquation 17 (and even Equation 6) is satisfied for the scalar functions givenin Equation 18. (cid:116)(cid:117) Theorem 3
Proof of Lemma 2
Proof of the Lemma is straightforward. Let ¯ L = ( ˜ L, L )and ¯ R = ( ˜ R, R ) be the restrictions of ¯Ψ on two adjacent mesh elements. ¯Ψagrees with degree m global regular in-plane parametrisation ˜Π, therefore ˜ L and ˜ R are polynomials of degree m . Conventional weight functions l ( v ), r ( v )and c ( v ) are evidently polynomials of (formal) degrees 2 m −
1, 2 m − m respectively. ¯Ψ ∈ ¯ FUN (( n )) and so R and L are polynomials of (formal) degree n . Using B´ezier representation of Z -components of the partial derivatives ¯ L u ,¯ R u , ¯ R v and of the weight functions l , r , cL u = n (cid:80) nj =0 ∆ L j B nj l = (cid:80) m − k =0 l k B m − k R u = n (cid:80) nj =0 ∆ R j B nj r = (cid:80) m − k =0 r k B m − k L v = n (cid:80) n − j =0 ∆ C j B n − j c = (cid:80) mk =0 c k B mk (109)one gets: L u ( v ) l ( v ) + R u ( v ) r ( v ) + L v ( v ) c ( v ) = n n +2 m − (cid:80) s =0 1 (cid:18) n+2m-1s (cid:19) (cid:80) j + k = s ≤ j ≤ n ≤ k ≤ m − (cid:16) nj (cid:17) (cid:16) (cid:17) ( l k ∆ L j + r k ∆ R j ) + (cid:80) j + k = s ≤ j ≤ n − ≤ k ≤ m (cid:16) n-1j (cid:17) (cid:16) (cid:17) c k ∆ C j B n +2 m − s ( v ) (110)Writing down equality to zero for every coefficient of the resulting B´ezierpolynomial leads to Equation 19.The number of necessary and sufficient equations is equal to the actual degreeof the polynomial L u ( v ) l ( v ) + R u ( v ) r ( v ) + L v ( v ) c ( v ), that is max { max deg ( l, r ) + n, deg ( c ) + n − } . (cid:116)(cid:117) Lemma 2
Proofs Proof of Lemma 7
It is sufficient to prove the Lemma for two adjacentpatches.Let two adjacent patches ¯ L and ¯ R be defined by a bilinear in-plane parametri-sation, the XY -components of the first and second order 3 D partial derivatives,computed at the common vertex (see Subsection 2.6.3), have the followingrepresentation in terms of the initial mesh data:¯ (cid:15) ( R ) = (˜ e ( R ) , n ∆ R ) ¯ τ ( R ) = (˜ t ( R ) , n (∆ R − ∆ R ))¯ (cid:15) ( L ) = (˜ e ( L ) , − n ∆ L ) ¯ τ ( L ) = (˜ t ( L ) , − n (∆ L − ∆ L ))¯ (cid:15) ( C ) = (˜ e ( C ) , n ∆ C ) (111)Here ˜ e ( R ) , ˜ e ( R ) , ˜ e ( R ) , ˜ e ( R ) , ˜ e ( R ) are directed planar edges and ¯ τ ( R ) , ¯ τ ( L ) are thetwist characteristics of the planar mesh elements (see Subsection 2.3.1).Let ”Eq(0)” be satisfied for the common edge of ¯ L and ¯ R . It is sufficient toshow that ”Eq(1)” is equivalent to the following equation tw ( R ) + tw ( L ) = coef f ( C ) δ ( C ) (112)where tw ( R ) = (cid:104) ˜ e ( R ) , ˜ e ( C ) (cid:105) mix ¯ τ ( R ) ¯ (cid:15) ( R ) ¯ (cid:15) ( C ) tw ( L ) = (cid:104) ˜ e ( C ) , ˜ e ( L ) (cid:105) mix ¯ τ ( L ) ¯ (cid:15) ( C ) ¯ (cid:15) ( L ) (113) coef f ( C ) = (cid:104) ˜ e ( R ) , ˜ e ( L ) (cid:105)(cid:104) ˜ e ( R ) , ˜ e ( C ) (cid:105)(cid:104) ˜ e ( C ) , ˜ e ( L ) (cid:105) (114)and δ ( C ) is Z -component of the second-order derivative along the common edge(see Subsection 2.6.3).Indeed, the following relations between coefficients of the weight functionsand geometrical characteristics of the planar mesh hold l = (cid:104) ˜ e ( R ) , ˜ e ( C ) (cid:105) , l − l = (cid:104) ˜ t ( R ) , ˜ e ( C ) (cid:105) r = −(cid:104) ˜ e ( C ) , ˜ e ( L ) (cid:105) , r − r = (cid:104) ˜ t ( L ) , ˜ e ( C ) (cid:105) (115)According to these formulas, the first two summands of ”Eq(1)” can be rewrittenas follows n ( l ∆ L + r ∆ R ) + ( l ∆ L + r ∆ R ) = n ( l (∆ L − ∆ L ) + r (∆ R − ∆ R )) + (cid:104) ˜ t ( R ) , ˜ e ( C ) (cid:105) ∆ L + (cid:104) ˜ t ( L ) , ˜ e ( C ) (cid:105) ∆ R +( n + 1)( l ∆ L + r ∆ R ) = n (cid:0) (cid:104) ˜ e ( C ) , ˜ e ( R ) (cid:105) τ ( L ) + (cid:104) ˜ t ( L ) , ˜ e ( C ) (cid:105) (cid:15) ( R ) (cid:1) − n (cid:0) (cid:104) ˜ e ( C ) , ˜ e ( L ) (cid:105) τ ( R ) + (cid:104) ˜ t ( R ) , ˜ e ( C ) (cid:105) (cid:15) ( L ) (cid:1) +( n + 1)( l ∆ L + r ∆ R ) = n mix ¯ τ ( L ) ¯ (cid:15) ( C ) ¯ (cid:15) ( R ) + (cid:104) ˜ t ( L ) , ˜ e ( R ) (cid:105) (cid:15) ( C ) − n mix ¯ τ ( R ) ¯ (cid:15) ( C ) ¯ (cid:15) ( L ) + (cid:104) ˜ t ( R ) , ˜ e ( L ) (cid:105) (cid:15) ( C ) +( n + 1)( l ∆ L + r ∆ R ) = n mix ¯ τ ( L ) ¯ (cid:15) ( C ) ¯ (cid:15) ( R ) − mix ¯ τ ( R ) ¯ (cid:15) ( C ) ¯ (cid:15) ( L ) + (cid:0) (cid:104) ˜ t ( L ) , ˜ e ( R ) (cid:105) − (cid:104) ˜ t ( R ) , ˜ e ( L ) (cid:105) (cid:1) ∆ C + ( n + 1)( l ∆ L + r ∆ R ) (116) Proofs The assumption that ”Eq(0)” is satisfied implies that l ∆ L + r ∆ R = − c ∆ C and so ”Eq(1)” is equivalent to the equation0 = n mix ¯ τ ( L ) ¯ (cid:15) ( C ) ¯ (cid:15) ( R ) − mix ¯ τ ( R ) ¯ (cid:15) ( C ) ¯ (cid:15) ( L ) + (cid:8) (cid:104) ˜ t ( L ) , ˜ e ( R ) (cid:105) − (cid:104) ˜ t ( R ) , ˜ e ( L ) (cid:105) + 2 c − ( n + 1) c (cid:9) ∆ C + ( n − c ∆ C (117)It can easily be shown, that (cid:104) ˜ t ( L ) , ˜ e ( R ) (cid:105) − (cid:104) ˜ t ( R ) , ˜ e ( L ) (cid:105) + 2 c = 2 (cid:104) ˜ ρ − ˜ γ, ˜ λ − ˜ γ (cid:105) = 2 (cid:104) ˜ e ( R ) , ˜ e ( L ) (cid:105) = 2 c (118)Therefore Equation 117 can be further simplified as follows0 = n mix ¯ τ ( L ) ¯ (cid:15) ( C ) ¯ (cid:15) ( R ) − mix ¯ τ ( R ) ¯ (cid:15) ( C ) ¯ (cid:15) ( L ) + ( n − c (∆ C − ∆ C ) = n mix ¯ τ ( L ) ¯ (cid:15) ( C ) ¯ (cid:15) ( R ) − mix ¯ τ ( R ) ¯ (cid:15) ( C ) ¯ (cid:15) ( L ) + (cid:104) ˜ e ( R ) , ˜ e ( L ) (cid:105) δ ( C ) (119)Now equation ”Eq(0)” is used in order to make the last equation more sym-metric. The tangent coplanarity in particular means that¯ (cid:15) ( R ) = (cid:104) ˜ e ( C ) , ˜ e ( L ) (cid:105) (cid:0) (cid:104) ˜ e ( R ) , ˜ e ( L ) (cid:105) ¯ (cid:15) ( C ) − (cid:104) ˜ e ( R ) , ˜ e ( C ) (cid:105) ¯ (cid:15) ( L ) (cid:1) ¯ (cid:15) ( L ) = (cid:104) ˜ e ( R ) , ˜ e ( C ) (cid:105) (cid:0) (cid:104) ˜ e ( R ) , ˜ e ( L ) (cid:105) ¯ (cid:15) ( C ) − (cid:104) ˜ e ( C ) , ˜ e ( L ) (cid:105) ¯ (cid:15) ( R ) (cid:1) (120)Substitution of these formulas in the mixed products of Equation 119 gives0 = (cid:104) ˜ e ( R ) , ˜ e ( C ) (cid:105)(cid:104) ˜ e ( C ) , ˜ e ( L ) (cid:105) mix ¯ τ ( L ) ¯ (cid:15) ( L ) ¯ (cid:15) ( C ) + (cid:104) ˜ e ( C ) , ˜ e ( L ) (cid:105)(cid:104) ˜ e ( R ) , ˜ e ( C ) (cid:105) mix ¯ τ ( R ) ¯ (cid:15) ( C ) ¯ (cid:15) ( R ) + (cid:104) ˜ e ( R ) , ˜ e ( L ) (cid:105) δ ( C ) (121)In order to complete the proof it only remains to divide all the members of thelast equation by (cid:104) ˜ e ( R ) , ˜ e ( C ) (cid:105)(cid:104) ˜ e ( C ) , ˜ e ( L ) (cid:105) which is not equal to zero due to thestrict convexity of the quadrilateral mesh elements. (cid:116)(cid:117) Lemma 7
Proof of Lemma 8
The strict convexity of the mesh elements implies thatany inner even vertex ˜ V has degree val ( V ) = 4 at least.Let coef f ( j ) = 0 for every j = 1 , . . . , val ( V ). In particular, coef f (2) = 0and so ˜ e (1) and ˜ e (3) are colinear and lie on some straight line ˜ l (1 , ; coef f (3) = 0and so ˜ e (2) and ˜ e (4) are colinear and lie on some straight line ˜ l (2 , (see Figure18). Therefore for val ( V ) = 4 the vertex is proven to be regular.It remains to show that val ( V ) could not be greater than 4. Indeed, let val ( V ) >
4. Then, ˜ e (2) should be colinear to both ˜ e (4) and ˜ e ( val ( V )) (becauseboth coef f (3) and coef f (1) are equal to zero). But ˜ e (4) and ˜ e ( val ( V )) can not be Proofs colinear because ˜ e ( val ( V )) lies strictly between ˜ e (4) and ˜ e ( deg (1)) which span anangle less than π due to the strict convexity of the mesh elements. (cid:116)(cid:117) Lemma 8
Proof of Lemma 9(1)
Compatibility of δ ( j ) ( j = 1 , . . . , val ( V )) with some second-order functionalderivatives in the functional sense means, that there exist such three scalars Z XX , Z XY , Z Y Y that the following relation holds for every j = 1 , . . . , val ( V ) δ ( j ) = ( α ( j ) , β ( j ) ) (cid:18) Z XX Z XY Z XY Z Y Y (cid:19) (cid:18) α ( j ) β ( j ) (cid:19) = (cid:0) α ( j ) (cid:1) Z XX + 2 α ( j ) β ( j ) Z XY + (cid:0) β ( j ) (cid:1) Z Y Y (122)where α ( j ) and β ( j ) are respectively X and Y -components of the planar edge˜ e ( j ) .In order to prove that the ”Circular Constraint” is satisfied one should provethat0 = val ( V ) (cid:80) j =1 ( − j (cid:104) ˜ e ( j − , ˜ e ( j +1) (cid:105)(cid:104) ˜ e ( j − , ˜ e ( j ) (cid:105)(cid:104) ˜ e ( j ) , ˜ e ( j +1) (cid:105) δ ( j ) = val ( V ) (cid:80) j =1 ( − j (cid:104) ˜ e ( j − , ˜ e ( j +1) (cid:105)(cid:104) ˜ e ( j − , ˜ e ( j ) (cid:105)(cid:104) ˜ e ( j ) , ˜ e ( j +1) (cid:105) (cid:110)(cid:0) α ( j ) (cid:1) Z XX + 2 α ( j ) β ( j ) Z XY + (cid:0) β ( j ) (cid:1) Z Y Y (cid:111) (123)independently of the values of Z XX , Z XY and Z Y Y . In other words it should beproven that the coefficient of every one of Z XX , Z XY and Z Y Y is equal to zero.Here it will be shown that the coefficient of Z XY is equal to zero; proof for thecoefficients of Z XX and Z Y Y can be given in the same way. Coefficient of Z XY is equal to2 val ( V ) (cid:88) j =1 ( − j η ( j ) (124)where η ( j ) = ( α ( j − β ( j +1) − α ( j +1) β ( j − ) α ( j ) β ( j ) ( α ( j − β ( j ) − α ( j ) β ( j − )( α ( j ) β ( j +1) − α ( j +1) β ( j ) ) (125)It can be shown by a straightforward computation that η ( j ) = 1 + ξ ( j − + ξ ( j ) , where ξ ( j ) = β ( j ) α ( j +1) α ( j ) β ( j +1) − α ( j +1) β ( j ) (126)The first statement of the Lemma immediately follows from the last equationbecause 2 (cid:80) val ( V ) j =1 ( − j η ( j ) = 2 (cid:80) val ( V ) j =1 ( − j (1 + ξ ( j − + ξ ( j ) ) is clearly equalto zero for an even value of val ( V ). (2) For a non-regular 4-vertex ˜ V at least one of the coefficients in the circu-lar equation ”Circular Constraint” is not equal to zero. Let us assume that Proofs coef f (4) (cid:54) = 0 and that δ (4) is defined as a dependent variable according to the ”Circular Constraint” . Then for any choice of δ (1) , δ (2) , δ (3) there exists such avalue of δ (4) that the ”Circular Constraint” is satisfied, and the value of δ (4) isuniquely defined.In order to prove the second statement of Lemma 9, one should show thatfor every choice of δ ( j ) ( j = 1 , . . . ,
4) which satisfies the ”Circular Constraint” ,there exist such scalars Z XX , Z XY , Z Y Y that Equation 122 holds for every j = 1 , . . . , δ (1) , δ (2) , δ (3) there exist such unique values of Z XX , Z XY , Z Y Y that Equation 122 holds for j = 1 , ,
3. In order to prove it,one can write the system of equation for δ (1) , δ (2) , δ (3) in the matrix form δ (1) δ (2) δ (3) = S Z XX Z XY Z Y Y , where S = (cid:0) α (1) (cid:1) α (1) β (1) (cid:0) β (1) (cid:1) (cid:0) α (2) (cid:1) α (2) β (2) (cid:0) β (2) (cid:1) (cid:0) α (3) (cid:1) α (3) β (3) (cid:0) β (3) (cid:1) (127)It is easy to show that det ( S ) = 2 (cid:104) ˜ e (1) , ˜ e (2) (cid:105)(cid:104) ˜ e (2) , ˜ e (3) (cid:105)(cid:104) ˜ e (3) , ˜ e (1) (cid:105) (cid:54) = 0 because (cid:104) ˜ e (1) , ˜ e (2) (cid:105) (cid:54) = 0, (cid:104) ˜ e (2) , ˜ e (3) (cid:105) (cid:54) = 0 as the vector products of consequent edges of thestrictly convex quadrilaterals and (cid:104) ˜ e (3) , ˜ e (1) (cid:105) (cid:54) = 0 according to the assumptionthat coef f (4) (cid:54) = 0. It means that for any choice of δ (1) , δ (2) , δ (3) values of Z XX , Z XY , Z Y Y are uniquely defined by equation Z XX Z XY Z Y Y = S − δ (1) δ (2) δ (3) (128)It remains to show that δ (4) also agrees with the second-order partial deriva-tives Z XX , Z XY , Z Y Y . The agreement holds because, there exists the uniquevalue of δ (4) so that the ”Circular Constraint” is satisfied, and from the firststatement of the Lemma it is already known that δ (4) = (cid:0) α (4) (cid:1) Z XX + 2 α (4) β (4) Z XY + (cid:0) β (4) (cid:1) Z Y Y satisfies the ”Circular Con-straint” . (cid:116)(cid:117) Lemma 9
Proof of Theorem 5
The sufficiently long proof of Theorem 5 starts with thetrivial Auxiliary Lemma 1. The Auxiliary Lemma just summarizes well-knownalgebraic facts, which will be useful in the proof.
Auxiliary Lemma 1:
Let Av = b be a linear system of equations, where A is p × q matrix with rows A , . . . , A p − , v is a vector of variables and b is a vector offree coefficients. A = − − A − −− − A − − ... − − A p − − − v = v v ... v q − b = b b ... b p − (129)Then Proofs (1) The system has a solution (is consistent) if and only if for every non-trivial set of coefficients α , . . . , α p − so that the linear combination ofrows of matrix A is equal to zero (cid:80) p − s =0 α s A s = 0, the correspondinglinear combination of the free coefficients is also equal to zero (cid:80) p − s =0 α s b s = 0. Here a ”non-trivial” set means that the set contains at least onenon-zero element. (2) The rank of the matrix A is equal to r if and only if for any set of coeffi-cients α , . . . , α p − so that (cid:80) p − s =0 α s A s = 0, the following two conditionshold (a) For any subset of l < p − r coefficients (cid:8) α s j (cid:9) lj =1 , equality of every oneof these coefficients to zero α s j = 0 ( j = 1 , . . . , l ) does not necessarilyimply that all other coefficients should also be equal to zero. (b) There exists such a subset of l = p − r coefficients (cid:8) α s j (cid:9) lj =1 that if α s j = 0 ( j = 1 , . . . , l ) then all other coefficients are necessarily equalto zero. (3) Under the assumption that q ≥ p the number of independent (free) vari-ables is equal to q − r . Which variables are free and which are dependentcan be defined, for example, by the Gauss elimination (pivoting) process.Now it is possible to proceed with a formal proof of Theorem 5. The proofis based on representation of the ”Middle” system of equations in the matrixform and uses a few Auxiliary Lemmas, which provide relations between a purealgebraic analysis of the system of equations and the geometrical essence oftheorem. (1) Consistency. It will be shown that for every inner edge, the ”Middle” system of equations is solvable in terms of the middle control points. Moreover,it will be shown that the ”Middle” system of equation has a solution in termsof the ”side” middle control points ( L t , R t ) for t = 2 , . . . , n − middle control points ˜ C t for t = 3 , . . . , n − middle control points ˜ C t for t = 3 , . . . , n − ”Middle” system in termsof control points ( L t , R t ) ( t = 2 , . . . , n −
2) is equivalent to solvability of thesystem in terms of differences ∆ L t , ∆ R t ( t = 2 , . . . , n − L t or R t .The proof of the solvability starts with the representation of the full systemof linearized G -continuity equations ”Eq(s)” ( s = 0 , . . . , n + 1) in the matrixform with respect to the differences ∆ L t , ∆ R t ( t = 0 , . . . , n ). Equation ”Eq(s)” ( s = 0 , . . . , n + 1) can be rewritten as follows( n + 1 − s )( l ∆ L s + r ∆ R s ) + s ( l ∆ L s − + r ∆ R s − ) + sumC s = 0 (130)where sumC s = ( n − s )( n +1 − s ) n c ∆ C s + s ( n +1 − s ) n c ∆ C s − + s ( s − n c ∆ C s − (131) Proofs For the full system of equations ”Eq(s)” ( s = 0 , . . . , n + 1), coefficients of ∆ L t and ∆ R t ( t = 0 , . . . , n ) form a ( n + 2) × n + 1) matrix A and expressions − sumC s ( s = 0 , . . . , n + 1) do not contain non-classified control points and areconsidered as the right-side coefficients b s = − sumC s ( s = 0 , . . . , n + 1). Onegets the following system, which corresponds to all n +2 linearized G -continuityequations along the given edge A ∆ L ∆ R ...∆ L n ∆ R n = b ... b n +1 (132)In the last system, equations for s = 0 , s = 1 , s = n, s = n + 1 are known to besatisfied and ∆ L t , ∆ R t for t = 0 , , n − , n do not contain any non-classifiedcontrol points.The ”Middle” system of equations with respect to the differences ∆ L t , ∆ R t ( t = 2 , . . . , n −
2) has the following representation in the matrix formˆ A ∆ L ∆ R ...∆ L n − ∆ R n − = ˆ b ...ˆ b n − (133)where- Matrix ˆ A is submatrix of matrix A , composed of rows s = 2 , . . . , n − L , ∆ R , . . . , ∆ L n − , ∆ R n − .- Right-side coefficients ˆ b , . . . , ˆ bn − b = b − (2 l ∆ L + 2 r ∆ R )ˆ b s = b s for s = 3 , . . . , n − , ˆ b n − = b n − − (2 l ∆ L n − + 2 r ∆ R n − ) (134)For example in the case of n = 4, matrix A has the form∆ L ∆ R ∆ L ∆ R ∆ L ∆ R ∆ L ∆ R ∆ L ∆ R s = 0 5 l r s = 1 l r l r s = 2 0 0 2 l r l r s = 3 0 0 0 0 3 l r l r s = 4 0 0 0 0 0 0 4 l r l r s = 5 0 0 0 0 0 0 0 0 5 l r (135) Proofs and ˆ A is the 2 × A ∆ L ∆ R s = 2 3 l r s = 3 3 l r (136)In the case of n = 5, A is a 7 ×
12 matrix and ˆ A is the 3 × L ∆ R ∆ L ∆ R s = 2 4 l r s = 3 3 l r l r s = 4 0 0 4 l r (137)In a general case matrix A of the full system and submatrix ˆ A of the ”Middle” system have the following structure (here horizontal and vertical lines separateelements which belong to submatrix ˆ A )The left upper corner of the matrix∆ L ∆ R ∆ L ∆ R ∆ L ∆ R ∆ L ∆ R s = 0 ( n +1) l ( n +1) r s = 1 l r nl nr s = 2 0 0 2 l r ( n − l ( n − r s = 3 0 0 0 0 3 l r ( n − l ( n − r The middle part of the matrix∆ L t − ∆ R t − ∆ L t ∆ R t s = t − n + 2 − t ) l ( n + 2 − t ) r s = t tl tr ( n + 1 − t ) l ( n + 1 − t ) r s = t +1 0 0 ( t + 1) l ( t + 1) r The right lower corner of the matrix∆ L n − ∆ R n − ∆ L n − ∆ R n − ∆ L n − ∆ R n − ∆ L n ∆ R n s = n − n − l ( n − r l r s = n − n − l ( n − r l r s = n nl nr l r s = n +1 0 0 0 0 0 0 ( n +1) l ( n +1) r The goal is to prove that the system defined by matrix ˆ A and the right-sidevector ˆ b is consistent. Let some non-trivial linear combination of the rows ofmatrix ˆ A be equal to zero (cid:80) n − s =2 α s ˆ A s = 0 In order to prove the consistency,one should verify that the corresponding linear combination of the right-sidecoefficients (cid:80) n − s =2 α s ˆ b s is also equal to zero. Auxiliary Lemma 2 provides thenecessary and sufficient conditions for equality to zero of a non-trivial linearcombination of the rows of matrix ˆ A . Auxiliary Lemma 2:
Let matrix ˆ A be the matrix of the ”Middle” system of equa-tions with respect to the differences ∆ L t , ∆ R t ( t = 2 , . . . , n −
2) (see Equation133). Then a non-trivial linear combination of its rows (cid:80) n − s =2 α s ˆ A s is equal tozero if and only if the following two conditions hold Proofs (1) The ”Projections Relation” is satisfied, or in other words, there exists sucha constant χ that χ = l l = r r (138) (2) Coefficients α s ( s = 2 , . . . , n −
1) are given by the formula α s = 2 n ( n + 1) (cid:16) n+1s (cid:17) ( − s χ s − α (139) Proof
See Appendix, after proof of Theorem 5.Now it remains to prove that if conditions (1) and (2) of Auxiliary Lemma2 hold then the linear combination of the right-side coefficients (cid:80) n − s =2 α s ˆ b s isequal to zero. Auxiliary Lemma 3 presents an important relation between thelinear combination, geometrical coefficient of proportionality χ defined by the ”Projections Relation” and coefficients of the weight function c ( v ). Auxiliary Lemma 3:
For an inner edge, let ”Eq(0)” -type and ”Eq(1)” -type equa-tions be satisfied, matrix ˆ A corresponds to the ”Middle” system of equationsand let a non-trivial linear combination of the rows of matrix ˆ A be equal to zero (cid:80) n − s =2 α s ˆ A s = 0. Then the corresponding linear combination of the right-sidecoefficients of the ”Middle” system has the following representation in terms ofconstant χ defined by ”Projections Relation” (see Equation 138) and coefficientsof the weight function c ( v ) n − (cid:88) s =2 α s ˆ b s = − nχ α (cid:40) n − (cid:88) s =0 ( − s (cid:16) n-1s (cid:17) χ s ∆ C s (cid:41) (cid:8) c − χc + χ c (cid:9) (140) Proof
See Appendix, after proof of Theorem 5.Auxiliary Lemma 4 contains the geometrical essence of the proof. It derivesthe relation between coefficients of the weight function c ( v ) and constant χ inthe case when the ”Projections Relation” is satisfied. Auxiliary Lemma 4:
For two adjacent mesh elements with bilinear in-plane parametri-sation, let the ”Projections Relation” hold and constant χ be defined by Equa-tion 138. Then the coefficients of the weight function c ( v ) satisfy the followingequation c − χc + χ c = 0 (141) Proof
See Appendix, after proof of Theorem 5.Auxiliary Lemma 4 clearly allows to complete the proof of the consistency of the ”Middle” system. Indeed, let a non-trivial linear combination (cid:80) n − s =2 α s ˆ A s be Proofs equal to zero. Then according to Auxiliary Lemma 2 the ”Projections Relation” holds and (cid:80) n − s =2 α s ˆ b s is equal to zero according to Auxiliary Lemmas 3 and 4. (2) Classification of the control points. In the previous part of the proofit was already shown that the ”Middle” system of equations is always solvablein terms of the ”side” middle control points L , . . . , L n − , R , . . . , R n − andthe ”central” middle control points ˜ C , . . . , ˜ C n − can be classified as basic(free)in advance. In addition, Auxiliary Lemma 2 shows that a non-trivial linearcombination of rows of matrix ˆ A (which corresponds to the ”Middle” system) isequal to zero if and only if the ”Projections Relation” holds and the coefficientsof the linear combination are defined by Equation 139.It implies that if the ”Projections Relation” does not hold , no non-triviallinear combination of the rows of matrix ˆ A is equal to zero and the matrix is ofthe full row rank r = n −
2. According to statement (3) of Auxiliary Lemma 1,the number of the basic (free) ”side” middle control points in this case is equalto 2( n − − ( n −
2) = n − middle controlpoints can be defined by the Gauss elimination process applied to the matrixˆ A . If the ”Projections Relation” holds , then rank r of the matrix ˆ A is equal to n −
3. Indeed, r ≤ n − A (see Equation 139). It remains to show that r ≥ n −
3. Onthe contrary, let r < n −
3. Then according to statement (2a) of AuxiliaryLemma 1, equality to zero of l = 1 coefficient in the zero linear combination ofrows (cid:80) n − s =2 α s ˆ A s = 0 should not necessarily imply that all other coefficients areequal to zero. However, it contradicts the recursive dependency (see Equation144) between coefficients of a non-trivial zero linear combination of the rows.The number of basic (free) ”side” middle control points in this case is equalto n − L j , ˜ R j ) for j = 2 , . . . , n −
2. Below theGauss elimination process is illustrated in the case of n = 5; in case of a general n the resulting matrix has precisely the same structure. For n = 5, applicationof two steps of the Gauss elimination process to matrix ˆ A of the ”Middle” systemof equations (see Equation 137) givesˆ A ˆ A ˆ A l r l r l r l r ˆ A ← ˆ A − χ ˆ A −→ l r l r l r ˆ A ← ˆ A − χ ˆ A −→ l r l r (142)The resulting matrix clearly shows that every second column contains a pivotand so in every pair ( ˜ L j , ˜ R j ) ( j = 2 ,
3) exactly one control point can be classifiedas a basic control point. In addition, for every pair ( ˜ L j , ˜ R j ) ( j = 2 , Proofs one. (cid:116)(cid:117) Theorem 5
Proof of Auxiliary Lemma 2
Conditions (1) and (2) are clearly sufficientin order to guarantee that the linear combination (cid:80) n − s =2 α s ˆ A s is equal to zero.It remains to show that the conditions are necessary. Indeed, equality (cid:80) n − s =2 α s ˆ A s = 0 implies that for columns which correspond to ∆ L t and ∆ R t ( t = 2 , . . . , n − α t ( n + 1 − t ) l + α t +1 ( t + 1) l = 0 α t ( n + 1 − t ) r + α t +1 ( t + 1) r = 0 (143)The existence of the constant χ (condition (1) ) immediately follows from thelast pair of equations and the assumption that the linear combination is non-trivial. In addition, Equation 143 defines the recursive dependency betweencoefficients of the linear combination α t +1 = − n + 1 − tt + 1 χα t , t = 2 , . . . , n − α s (Equation 139),which means that condition (2) is satisfied. (cid:116)(cid:117) AuxiliaryLemma 2
Proof of Auxiliary Lemma 3
The proof consists of two parts. The first partshows that the linear combination of the right-side coefficients of the ”Middle” system of equations (cid:80) n − s =2 α s ˆ b s is equal to a similar linear combination of theright-side coefficients of the full system of equations (cid:80) n +1 s =0 α s b s . The secondpart expresses the linear combination (cid:80) n +1 s =0 α s b s in terms of χ and coefficientsof the weight function c ( v ).Auxiliary Lemma 2 implies, that the ”Projections Relation” holds and con-stant χ is correctly defined. According to the definition of ˆ b (see Equation134), definition of χ and the assumption that equations ”Eq(0)” and ”Eq(1)” are satisfied, one getsˆ b = b − l ∆ L + r ∆ R ) ( definition of χ ) = b − χ − ( l ∆ L + r ∆ R ) ” Eq ( )” = b − n χ − ( b − n +1 ( l ∆ L + r ∆ R )) ( definition of χ ) = b − n χ − ( b − n +1 χ − ( l ∆ L + r ∆ R )) ” Eq ( )” = b − n χ − b + n ( n +1) χ − b (145)In the same manner, it can be shown thatˆ b n − = b n − − n χb n + 2 n ( n + 1) χ b n +1 (146) Proofs Let α s for s = 0 , , n, n +1 be formally defined according to the recursive relation144 α = n ( n +1) χ − α , α = − n χ − α α n = − n χα n − , α n +1 = n ( n +1) χ α n − (147)Then Equations 145, 146 allow to conclude that n − (cid:88) s =2 α s ˆ b s = n +1 (cid:88) s =0 α s b s (148)Now it remains to substitute the explicit formulas for α s (see AuxiliaryLemma 2, Equation 139) and b s = − sumC s (see Equation 131) into the rightside of Equation 148. A straightforward computation of (cid:80) n +1 s =0 α s b s gives n +1 (cid:80) s =0 α s b s = − n +1 (cid:80) s =0 α s sumC s = − n n +1 (cid:80) s =0 α s { ( n − s )( n +1 − s ) c ∆ C s +2 s ( n +1 − s ) c ∆ C s − + s ( s − c ∆ C s − } = − n n − (cid:80) s =0 ∆ C s { α s c ( n − s )( n +1 − s )+2 α s +1 c ( s +1)( n − s )+ α s +2 c ( s +1)( s +2) } = − n α χ (cid:26) n − (cid:80) s =0 ( − s (cid:16) n-1s (cid:17) χ s ∆ C s (cid:27) (cid:8) c − χc + χ c (cid:9) (149) Equation 149 completes the proof of Auxiliary Lemma 3. (cid:116)(cid:117)
AuxiliaryLemma 3
Proof of Auxiliary Lemma 4
Let ϕ, ϕ (cid:48) , ψ, ψ (cid:48) be angles between the common edge of two patches and leftand right adjacent edges (see Figure 21). In the case of the bilinear in-planeparametrisation, coefficients of the weight functions l ( v ) and r ( v ) have the fol-lowing representations in terms of the geometrical characteristics of two adjacentmesh elements (see Equation 29) l = (cid:104) ˜ ρ − ˜ γ, ˜ γ (cid:48) − ˜ γ (cid:105) = || ˜ γ (cid:48) − ˜ γ || || ˜ ρ − ˜ γ || sinψl = (cid:104) ˜ γ − ˜ γ (cid:48) , ˜ ρ (cid:48) − ˜ γ (cid:48) (cid:105) = || ˜ γ (cid:48) − ˜ γ || || ˜ ρ (cid:48) − ˜ γ (cid:48) || sinψ (cid:48) r = −(cid:104) ˜ γ (cid:48) − ˜ γ, ˜ λ − ˜ γ (cid:105) = −|| ˜ γ (cid:48) − ˜ γ || || ˜ λ − ˜ γ || sinϕr = −(cid:104) ˜ λ (cid:48) − ˜ γ (cid:48) , ˜ γ − ˜ γ (cid:48) (cid:105) = −|| ˜ γ (cid:48) − ˜ γ || || ˜ λ (cid:48) − ˜ γ (cid:48) || sinϕ (cid:48) (150)Then the ”Projections Relation” implies that χ = || ˜ ρ − ˜ γ || sinψ || ˜ ρ (cid:48) − ˜ γ (cid:48) || sinψ (cid:48) = || ˜ λ − ˜ γ || sinϕ || ˜ λ (cid:48) − ˜ γ (cid:48) || sinϕ (cid:48) (151)For coefficients of the weight function c ( v ) the following relations hold c = (cid:104) ˜ ρ − ˜ γ, ˜ λ − ˜ γ (cid:105) = || ˜ ρ − ˜ γ || || ˜ λ − ˜ γ || sin ( ϕ + ψ ) c = −(cid:104) ˜ λ (cid:48) − ˜ γ (cid:48) , ˜ ρ (cid:48) − ˜ γ (cid:48) (cid:105) = −|| ˜ ρ (cid:48) − ˜ γ (cid:48) || || ˜ λ (cid:48) − ˜ γ (cid:48) || sin ( ϕ (cid:48) + ψ (cid:48) ) c = (cid:16) (cid:104) ˜ ρ − ˜ γ, ˜ λ (cid:48) − ˜ γ (cid:48) (cid:105) − (cid:104) ˜ λ − ˜ γ, ˜ ρ (cid:48) − ˜ γ (cid:48) (cid:105) (cid:17) = (cid:16) || ˜ ρ − ˜ γ || || ˜ λ (cid:48) − ˜ γ (cid:48) || sin ( ϕ (cid:48) − ψ ) − || ˜ ρ (cid:48) − ˜ γ (cid:48) || || ˜ λ − ˜ γ || sin ( ϕ − ψ (cid:48) ) (cid:17) (152) Proofs Using the standard trigonometric formulas and Equation 151, it is possibleto rewrite 2 χc in the form which clearly shows that Auxiliary Lemma 4 issatisfied2 χc = χ || ˜ ρ − ˜ γ || || ˜ λ (cid:48) − ˜ γ (cid:48) || ( sinϕ (cid:48) cosψ − sinψ cosϕ (cid:48) ) − χ || ˜ ρ (cid:48) − ˜ γ (cid:48) || || ˜ λ − ˜ γ || ( sinϕ cosψ (cid:48) − sinψ (cid:48) cosϕ ) = || ˜ ρ − ˜ γ || || ˜ λ − ˜ γ || ( sinϕ cosψ + sinψ cosϕ ) − χ || ˜ ρ (cid:48) − ˜ γ (cid:48) || || ˜ λ (cid:48) − ˜ γ (cid:48) || ( sinψ (cid:48) cosϕ (cid:48) + sinϕ (cid:48) cosψ (cid:48) ) = c + χ c (153) (cid:116)(cid:117) Auxiliary Lemma 4
Proof of Lemma 11(1)
It should be shown that the direction of every mesh edge is defined mostlyone connected component of the D -dependency graph (by mostly one D -dependencytree). A mesh edge ˜ e is directed if it is an edge of the spanning tree of someconnected component ˜ C of the D -dependency graph or if it corresponds to adangling half-edge of the root vertex of ˜ C . In the first case the direction of ˜ e isobviously uniquely defined, because ˜ e connects two vertices of the same compo-nent and can not be influenced by any other component. In the second case ˜ e clearly could not be an edge of the spanning tree of some other component ˜ C (cid:48) (otherwise it would connect two vertices of ˜ C (cid:48) ). The only situation when compo-nent ˜ C (cid:48) may affect the orientation of ˜ e is the situation when ˜ e also correspondsto a dangling half-edge of the root vertex of ˜ C (cid:48) . But it is impossible because inthis case roots of ˜ C and ˜ C (cid:48) should belong to the same connected component ofthe D -dependency graph due to the connection by ˜ e . (2) On the contrary, let some vertex ˜ V use the D -type control point of directededge ˜ e = ( ˜ V , ˜ V (cid:48) ) and let ˜ V belong to D -dependency tree T while ˜ e belongsto D -dependency tree T (cid:48) . The only situation when directed edge ˜ e does notbelong to the same D -dependency tree as its vertex ˜ V is the situation when˜ V (cid:48) is the root vertex of T (cid:48) and ˜ e corresponds to a dangling half-edge of ˜ V (cid:48) .Then according to the definition of the dangling half-edge it means that ˜ V (cid:48) does not use the D -type control point of ˜ e , which leads to a contradiction withthe assumption. (3) The third statement of Lemma 11 is an obvious result of Algorithms 2 and3 for the construction of the D -dependency graph and of the D -dependencyforest. (cid:116)(cid:117) Lemma 11
Proof of Theorem 6
Let ˜ C be a connected component of the D -dependency graph. The onlytype of primary vertices which may belong to the component is the inner evenvertices, excluding regular 4-regular vertices, because no other vertex uses any D -type control points according to the assumption of theorem. Let ˜ R be the Proofs rightmost (geometrically) primary vertex which belongs to ˜ C or the lowest right-most vertex in case of ambiguity. It will be shown that ˜ R has at least onedangling half-edge.Let ˜ r be a vertical line which passes through ˜ R . Then in the initial meshthere exist edges ( ˜ R, ˜ V (1) ) , . . . , ( ˜ R, ˜ V ( k ) ) ( k ≥
1) which lie strictly on the rightfrom ˜ r or at the lower half of ˜ r (see Figure 54(a)). Indeed, ˜ R is an inner vertexand so it is surrounded by domain ˜Ω. If all mesh edges adjacent to ˜ R lie onthe left of ˜ r or on the upper half of ˜ r , there should exist a quadrilateral meshelement with inner angle ≥ π , which contradicts the strict convexity of meshelements (see Figure 54(b)).Edges ( ˜ R, ˜ V (1) ) , . . . , ( ˜ R, ˜ V ( k ) ) do not belong to the D -dependency graph,otherwise vertices ˜ V (1) , . . . , ˜ V ( k ) would belong to ˜ C , which contradicts the factthat ˜ R is the lowest rightmost primary vertex of ˜ C . It means that for everyone of edges ( ˜ R, ˜ V ( j ) ) ( j = 1 , . . . , k ) at least one of its vertices does not use the D -type control point ˜ D ( j ) corresponding to the edge. The purpose is to show,that ˜ R uses the D -type control point of at least one of these edges; in this case˜ R and the secondary vertex of this edge form a dangling half-edge.On the contrary, let ˜ R do not use any ˜ D ( j ) ( j = 1 , . . . , k ).Let k = 1 and ˜ R does not use ˜ D (1) . Then in the initial mesh ( ˜ R, ˜ W (1) ) and( ˜ R, ˜ W (2) ) - two neighboring edges of ( ˜ R, ˜ V (1) ) - should be colinear (see Figure54(c)). In this case one of the vertices W (1) , W (2) lies strictly on the right from˜ r or on the lower half of ˜ r , which contradicts the assumption that k = 1.Let k ≥
3. Then angle ∠ ˜ V (1) ˜ R ˜ V ( k ) < π (see Figure 54(d)) and for any j = 2 , ..., k − D -type control point of ( ˜ R, ˜ V ( j ) ) should participate in the ”Circular Constraint” of the vertex ˜ R with a non-zero coefficient. It means that˜ R uses ˜ D ( j ) for j = 2 , . . . , k − k = 2. If ˜ R does not use D -type control points of both( ˜ R, ˜ V ((1) ) and ( ˜ R, ˜ V (2) ), then in the initial mesh there should exist two suchpairs of colinear edges that the edges form a continuous sequence with respectto the order of edges around ˜ R . For example in Figure 54(e), ˜ V (1) , ˜ R, ˜ W (1) and˜ V (2) , ˜ R, ˜ W (2) are triples of the colinear points, and edges ( ˜ R, ˜ W (2) ), ( ˜ R, ˜ V (1) ),( ˜ R, ˜ V (2) ), ( ˜ R, ˜ W (1) ) follow one another in the counter-clockwise order around ˜ R .If deg ( R ) = 4, then ˜ R should be a regular 4-vertex, which contradicts the factthat ˜ R belongs to the D -dependency graph. If deg ( R ) ≥ R contradicts the ”Uniform Edge Distribution Condition” which is assumed to be satisfied. (cid:116)(cid:117) Theorem 6
Proof of Theorem 7
Let |B ( n ) V,E − type | , |B ( n ) D,T − type | , |B ( n ) middle | denote respectively the number of basiccontrol points of V , E -type, D , T -type and of the middle control points whichparticipate in G -continuity conditions. Then | ˜ B ( n ) G | = |B ( n ) V,E − type | + |B ( n ) D,T − type | + |B ( n ) middle | (154) Proofs Clearly, for any n |B ( n ) V,E − type | = 3 | V ert non − corner | (155)In addition, the following useful equality always holds (cid:80) ˜ V non − corner val ( V ) = 2 (cid:34) | Edge inner | + | V ert boundarynon − corner | (cid:35) (156)Let n ≥
5. Then local templates for classification of D , T -type control pointsnever intersect and the number of basic D , T -type control points adjacent toa non-corner vertex ˜ V is equal to val ( V ) + 1 for inner regular 4-vertices, to2 = val ( V ) − val ( V ) for the rest of vertices.Therefore |B ( n ) D,T − type | = (cid:80) ˜ V non − corner val ( V ) + | V ert inner − regular | − | V ert boundarynon − corner | (157)According to Theorem 5, |B ( n ) middle | = (2 n − | Edge inner | + | Edge inner, ” P rojections Relation ” holds | (158)Equations 155, 157, 158 and 156 allow to conclude that the formula for | ˜ B ( n ) G | ,given in Theorem 7, is correct for n ≥ n = 4. Then, according to Theorem 5, |B (4) middle | = | Edge inner, ” P rojections Relation ” holds | (159)The number of basic D -type and T -type control points can be computed asfollows. The total number of D -type control points which belong to ˜ B (4) G isequal to | Edge inner | . One dependent D -type control point is assigned to every D -relevant vertex. Therefore |B (4) D − type | = | Edge inner | − |
V ert inner, even,not − regular | − | V ert boundaryD − relevant | (160)The number of basic T -type control points is given by |B (4) T − type | = | V ert inner, even,not − regular | + | V ert inner, even, − regular | + | V ert boundary, non − corner,not D − relevant | + 2 | V ert boundaryD − relevant | (161)Therefore the number of D -type and T -type basic control point together donot depend on the presence of the boundary D -relevant vertices and can becomputed by the formula |B (4) D,T − type | = | Edge inner | − |
V ert inner − regular | − | V ert boundarynon − corner | (162) Proofs From Equations 155, 157, 158 it follows that | ˜ B (4) G | = 3 | V ert non − corner | + | V ert boundarynon − corner | + | Edge inner | + | V ert inner − regular | + | Edge inner, ” P rojections Relation ” holds | (163)The last expression fits the general formula (Equation 52) for n = 4. (cid:116)(cid:117) Theorem 7
Proof of Lemma 16
Lemma 16 has a very simple formal proof.If max deg ( l, r ) = 2 then the statement of the Lemma is evidently satisfiedbecause deg ( c ) ≤ max deg ( l, r ) = 1 then l ( power )2 = (cid:104) ˜ ρ ( power )2 , ˜ γ (cid:48) − ˜ γ (cid:105) = 0 and r ( power )2 = −(cid:104) ˜ λ ( power )2 , ˜ γ (cid:48) − ˜ γ (cid:105) = 0. It means that vectors ˜ λ ( power )2 , ˜ ρ ( power )2 , ˜ γ (cid:48) − ˜ γ areparallel and so c ( power )4 = (cid:104) λ ( power )2 , ρ ( power )2 (cid:105) = 0.If max deg ( l, r ) = 0 then in the same manner as in the previous case, onesees that vectors ˜ λ ( power )1 , ˜ ρ ( power )1 , ˜ λ ( power )2 , ˜ ρ ( power )2 , ˜ γ (cid:48) − ˜ γ are parallel and so c ( power )4 = c ( power )3 = 0. (cid:116)(cid:117) Lemma 16
Proof of Lemma 19
The system of n + 4 linear equations given in Lemma 18 has the followingrepresentation in terms of sumLR s and sumC s (see Definition 20) sumLR s + sumLR s − + sumC s = 0 (164)where s = 0 , . . . , n + 3.Expressing recursively sumLR s − from the previous equations for s = 0 , . . . , n +2 and leaving equation for s = n + 3 unchanged, one gets an equivalent system( s = 0) sumLR = − sumC ( s = 1) sumLR = − sumLR − sumC = sumC − sumC ...( s = n + 2) sumLR n +2 = ( − n +3 (cid:80) n +2 k =0 ( − sumC k ( s = n + 3) sumLR n +2 = − sumC n +3 (165)For s = 0 , . . . , n + 2, the equation has precisely the form of the indexed equation Eq(s) and it remains to note that the last couple of equations is equivalent tothe couple ”Eq(n+2)” and ”sumC-equation” . (cid:116)(cid:117) Lemma 19
Proof of Lemma 21
Proofs According to Lemma 17, statements of Lemma 21 clearly hold for aninner vertex which has no adjacent edges with one boundary vertex. Moreover, ”Eq(0)” -type and ”Eq(1)” -type equations remain unchanged for any edge withtwo inner verticesIt remains to consider equations ”Eq(0)” and ”Eq(1)” of an edge with oneboundary vertex, applied at the inner vertex of the edge.In order to distinguish between the case of ˜Π ( bilinear ) and the case of ˜Π ( bicubic ) global in-plane parametrisations, superscripts ( bilinear ) and ( bicubic ) are added tocoefficients of the weight functions and to the indexed equations. It is easy toverify (see Equations 29, 98, 99) that l ( bicubic )0 = l bilinear l ( bicubic )1 = ( l ( bilinear )0 + l ( bilinear )1 ) r ( bicubic )0 = r ( bilinear )0 r ( bicubic )1 = ( r ( bilinear )0 + r ( bilinear )1 ) c ( bicubic )0 = l ( bilinear )0 c ( bicubic )1 = ( c ( bilinear )0 + c ( bilinear )1 ) (166) (1) ” Eq (0) ( bicubic ) ” involves only zero-indexed coefficients of the weight func-tions, which are equal for global in-plane parametrisations ˜Π ( bilinear ) and ˜Π ( bicubic ) . (2) From Equation 166 and formulas for ”Eq(0)” and ”Eq(1)” in the case ofbilinear and in the case of bicubic in-plane parametrisations (see Equations 31,35 and 63) it follows that” Eq (0) ( bicubic ) ” = ” Eq (0) ( bilinear ) ”” Eq (1) ( bicubic ) ” = ” Eq (0) ( bilinear ) ” + ” Eq (1) ( bilinear ) ” (167)Therefore systems of equations ”Eq(0)” , ”Eq(1)” are equivalent for ˜Π ( bilinear ) and ˜Π( bicubic ) in-plane parametrisations. (cid:116)(cid:117) Lemma 21
Proof of Theorem 8 and Theorem 9
General approach to the proof.
By definition, the ”Middle” system of equa-tions consists of the ”sumC-equation” and the ”Restricted Middle” system of in-dexed equations. ”sumC-equation” involves the ”central” middle control pointsalone, while every one of the indexed equations ”Eq(s)” involves both the ”cen-tral” and the ”side” middle control points.One of the basic concepts of the proof is the assumption that the ”central” middle control points are classified prior to and independently of classificationof the ”side” middle control points. The classification of the ”central” middle control points is assumed to be made is such a manner that ”sumC-equation” (and some additional requirements explained below) are satisfied. The assump-tion allows to consider the ”side” middle control points as the only variablesof the ”Restricted Middle” system , while the ”central” middle control pointsparticipate in the system as components of the right-side coefficients. A consis-tency analysis of the ”Restricted Middle” system establishes some requirements
Proofs which should be satisfied by the right-side coefficients of the system. The anal-ysis leads to the necessary and sufficient conditions formulated in Theorem 8,which define additional requirements to the classification of the ”central” middle control points.Note, that according to this approach, responsibilities of the ”central” andthe ”side” middle control points are shared in the following manner. The”central” middle control points are responsible for the satisfaction of ”sumC-equation” as well as for the consistency of the ”Restricted Middle” system ofequations, while the classification of the ”side” middle control points reflectsthe rank analysis of the ”Restricted Middle” system. Structure of the ”Restricted Middle” system of equations.
As it was men-tioned above, the proof starts with the consistency and rank analysis of the ”Restricted Middle” system of equations.Similarly to the case of global bilinear in-plane parametrisation ˜Π ( bilinear ) ,let A be matrix of the full system of linearized G -continuity equations excluding”sumC-equation” in terms of differences ∆ L t , ∆ R t t = 0 , . . . , n and let b s ( s =0 , . . . , n + 2) be the right-side coefficients of the system. In other words, matrix A and vector b correspond to the system of n + 3 indexed equations ”Eq(s)” s = 0 , . . . , n + 2 A ∆ L ∆ R ...∆ L n ∆ R n = b ... b n +2 (168)According to Lemma 19, A contains coefficients of ∆ L t , ∆ R t ( t = 0 , . . . , n ) inexpressions for sumLR s (see Definition 20) and b s = ( − s +1 s (cid:88) k =0 ( − k sumC k (169)for s = 0 , . . . , n + 2.Assumptions of Theorems 8 and 9 mean thatEquations ”Eq(s)” for s = 0 , s = 1 , s = n (cid:48) + 1 for n ≥ ”Eq(n’+2)” for s ≥ L t , ∆ R t for t = 0 , , n (cid:48) − n ≥ t = n (cid:48) for n ≥ A denote matrix of the ”Restricted Middle” system of equations in terms ofdifferences ∆ L t , ∆ R t ( t = 2 , . . . , n (cid:48) − A s ( s = 2 , . . . , n (cid:48) ) denotes a row of thematrix ˆ A and let ˆ b s ( s = 2 , . . . , n (cid:48) ) be the right-side coefficients of the system.Matrix ˆ A is a ( n (cid:48) − × n (cid:48) −
3) submatrix of matrix A composed of rows A s Proofs for s = 2 , . . . , n (cid:48) and columns corresponding to ∆ L , ∆ R , . . . , ∆ L n (cid:48) − , ∆ R n (cid:48) − .Matrix ˆ A has the following form ∆ L ∆ R ∆ L ∆ R . . . ∆ L n (cid:48) − ∆ R n (cid:48) − ∆ L n (cid:48) − ∆ R n (cid:48) − s = 2 ( n2 ) l ( n2 ) r . . . s = 3 2( n2 ) l n2 ) r ( n3 ) l ( n3 ) r . . . s = 4 ( n2 ) l ( n2 ) r n3 ) l n3 ) r . . . s = 5 0 0 ( n3 ) l ( n3 ) r . . . s = n (cid:48) − . . . ( nn’-3 ) l ( nn’-3 ) r s = n (cid:48) − . . . nn’-3 ) l nn’-3 ) r ( nn’-2 ) l ( nn’-2 ) r s = n (cid:48) − . . . ( nn’-3 ) l ( nn’-3 ) r nn’-2 ) l nn’-2 ) r s = n (cid:48) . . . nn’-2 ) l ( nn’-2 ) r (170)Right-side coefficients ˆ b s ( s = 2 , . . . , n (cid:48) ) are given by the next formula ˆ b = b − (cid:104)(cid:16) n0 (cid:17) l ∆ L + (cid:16) n0 (cid:17) r ∆ R + 2 (cid:16) n1 (cid:17) l ∆ L + 2 (cid:16) n1 (cid:17) r ∆ R (cid:105) ˆ b = b − (cid:104)(cid:16) n1 (cid:17) l ∆ L + (cid:16) n1 (cid:17) r ∆ R (cid:105) ˆ b s = b s for s = 4 , . . . , n (cid:48) − b n (cid:48) − = b n (cid:48) − − (cid:104)(cid:16) nn’-1 (cid:17) l ∆ L n (cid:48) − + (cid:16) nn’-1 (cid:17) r ∆ R n (cid:48) − (cid:105) ˆ b n (cid:48) = b n (cid:48) − (cid:104) (cid:16) nn’-1 (cid:17) l ∆ L n (cid:48) − +2 (cid:16) nn’-1 (cid:17) r ∆ R n (cid:48) − + (cid:16) nn’ (cid:17) l ∆ L n (cid:48) + (cid:16) nn’ (cid:17) r ∆ R n (cid:48) (cid:105) (171)For example, for n = 4 and n = 5, matrix A has the explicit form givenbelow, where the last row and two last columns are relevant only in the case of n = 5 (the row and column are separated by single lines). ∆ L ∆ R ∆ L ∆ R ∆ L ∆ R ∆ L ∆ R ∆ L ∆ R ∆ L ∆ R s =0 ( n0 ) l ( n0 ) r s =1 2( n0 ) l n0 ) r ( n1 ) l ( n1 ) r s =2 ( n0 ) l ( n0 ) r n1 ) l n1 ) r ( n2 ) l ( n2 ) r s =3 0 0 ( n1 ) l ( n1 ) r n2 ) l n2 ) r ( n3 ) l ( n3 ) r s =4 0 0 0 0 ( n2 ) l ( n2 ) r n3 ) l n3 ) r ( n4 ) l ( n4 ) r s =5 0 0 0 0 0 0 ( n3 ) l ( n3 ) r n4 ) l n4 ) r ( n5 ) l ( n5 ) r s =6 0 0 0 0 0 0 0 0 ( n4 ) l ( n4 ) r n5 ) l n5 ) r s =7 0 0 0 0 0 0 0 0 0 0 ( n5 ) l ( n5 ) r For both n = 4 and n = 5 matrix ˆ A is a middle 4 × A , composedof rows s = 2 , , , L , ∆ R , ∆ L , ∆ R ( ˆ A isseparated by double lines). The possible types of dependencies between coefficients of conventionalweight functions l ( v ) and r ( v ) . As shown in the previous paragraph, matrixˆ A contains coefficients of conventional weight functions l ( v ) and r ( v ). Differ-ent types of dependencies between coefficients of the weight functions lead tothe different consistency and rank analysis of the ”Restricted Middle” systemof equations. Definition 25 presents a refinement of the possible types of de-pendencies listed in Theorem 9. Subdivision into these subcases and auxiliary Proofs notations given in the Definition are justified by the series of the following Aux-iliary Lemmas. Definition 25:
Let conventional weight functions l ( v ) and r ( v ) be defined byglobal in-plane parametrisation ˜Π ( bicubic ) . Then for an edge with one boundaryvertex, the following types of dependencies between coefficients of the weightfunctions l ( v ) and r ( v ) will be considered (here g ( ij ) , i, j ∈ { , , } are givenby Definition 22) • ”CASE 0” complement of ”CASE 1” and ”CASE 2” • ”CASE 1” g (01) g (12) g (02) (cid:54) = 0 , (cid:110) g (02) (cid:111) = 4 g (01) g (12) In this case it is possible to define such a scalar value χ that χ = − g (01) g (02) = − g (02) g (12) (172)The following two subcases are defined ”CASE 1.a” χ (cid:54) = − ”CASE 1.b” χ = − • ”CASE 2” g (01) = g (12) = g (02) = 0In this case it is possible to define such scalars χ ( ij ) ( i, j ∈ { , , } ) that χ ( ij ) = l i l j = r i r j (173)The special notations are used for χ (02) and χ (12) ξ = χ (02) , η = χ (12) (174)and matrix G is defined as follows G = (cid:18) − ξ ηξ − η η − ξ (cid:19) (175) ”CASE 2.1” η (cid:54) = ξ In this case G has two different eigenvalues ω and ω ”CASE 2.1.a” √ ω = η + (cid:112) η − ξ (cid:54) = 1 and √ ω = η − (cid:112) η − ξ (cid:54) = 1 (176) ”CASE 2.1.b” √ ω = 1 or √ ω = 1 ”CASE 2.2” η = ξ In this case G has a single eigenvalue ω ”CASE 2.2.a” √ ω = η (cid:54) = 1 ”CASE 2.2.b” √ ω = 1 Proofs Explanation regarding the subdivision into the basic cases ”CASE 0” , ”CASE1” and ”CASE 2” . Both proof of the consistency and the rank analysis of ”Restricted Middle” system deal with zero non-trivial linear combinations ofrows. Auxiliary Lemma 5 provides necessary and sufficient conditions for theexistence of such a linear combination and describes the structure of the setof its coefficients. Subdivision into three basic cases ”CASE 0” , ”CASE1” , ”CASE 2” directly follows from the classification given in the AuxiliaryLemma. Auxiliary Lemma 5:
A non-trivial combination of the rows of matrix ˆ A is equalto zero n (cid:48) (cid:88) s =2 α s ˆ A s = 0 (177)if and only if either one of the following conditions holds (1) Conditions of ”CASE 1” are satisfied andCoefficients of the linear combination obey the recursive relation α s +1 = χα s , s = 2 , . . . , n (cid:48) − (2) Conditions of ”CASE 2” are satisfied andCoefficients of the linear combination obey the recursive relation α s +2 = − ξα s − ηα s +1 , s = 2 , . . . , n (cid:48) − Proof
See Appendix, after proof of Theorem 8 and Theorem 9.
Conclusion 3:
Auxiliary Lemma 5 implies that the rank and consistency of the ”Restricted Middle” system of equations depend in the following way on thesubdivision in the basic cases ”CASE 0”
Matrix ˆ A has the full row rank, rank ( ˆ A ) = n (cid:48) − ”Restricted Middle” system is consistent. ”CASE 1” Matrix ˆ A has row rank deficiency 1, rank ( ˆ A ) = n (cid:48) − ”Restricted Middle” system is consistent if and only if for any setof coefficients { α a } n (cid:48) s =2 which satisfy Equation 178, the correspondinglinear combination of the right sides is equal to zero (cid:80) n (cid:48) s =2 α s ˆ b s = 0. ”CASE 2” Matrix ˆ A has row rank deficiency 2, rank ( ˆ A ) = n (cid:48) − ”Restricted Middle” system is consistent if and only if for any setof coefficients { α a } n (cid:48) s =2 which satisfy Equation 179, the correspondinglinear combination of the right sides is equal to zero (cid:80) n (cid:48) s =2 α s ˆ b s = 0. Proofs Conclusion 3 allows to study the necessary and sufficient conditions for theconsistency of the ”Restricted Middle” system in terms of right-side coefficients,or in other words, in terms of the ”central” middle control points.Before proceeding with the consistency analysis, let us give some additionalexplanations concerning subdivision of ”CASE 2” , which has the most com-plicated structure.
Explanation regarding the subdivision of ”CASE 2” into ”CASE 2.1” and ”CASE 2.2” . Auxiliary Lemma 6 justifies the definition of matrix G intro-duced in Equation 175. Auxiliary Lemma 6:
Let conditions of ”CASE 2” be satisfied and let matrix G be defined by Equation 175. Let further { α s } n (cid:48) s =2 be coefficients of a non-trivialzero combination of rows (cid:80) n (cid:48) s =2 α s ˆ A s = 0. Then dependency of the coefficientshas the following matrix form( α s +2 , α s +3 ) = ( α s , α s +1 ) G (180)for s = 2 , . . . , n (cid:48) − Proof
Auxiliary Lemma 6 immediately follows from Equation 179 given inAuxiliary Lemma 5. (cid:116)(cid:117)
Auxiliary Lemma 6
Auxiliary Lemma 7 describes the Jordan form of matrix G . The Jordanform appears to play an important role in the consistency analysis and explainsthe subdivision of ”CASE 2” into subcases ”CASE 2.1” and ”CASE 2.2” . Auxiliary Lemma 7:
Let the conditions of ”CASE 2” be satisfied. Then thematrix G (see Equation 175) has the following eigenvalues, eigenvectors andJordan form ”CASE 2.1” Matrix G has two different eigenvalues ω , = − ξ + 2 η ( η ± (cid:112) η − ξ ) (181)which correspond to the eigenvectors v , = (cid:18) ξη ± (cid:112) η − ξ (cid:19) = (cid:18) ξ √ ω , (cid:19) (182)and G can be represented in the form G = ( v v ) (cid:18) ω ω (cid:19) ( v v ) − (183) Proofs ”CASE 2.2” Matrix G has a single eigenvalue ω = η (184)which corresponds to the eigenvector v = (cid:18) η η (cid:19) = (cid:18) ω √ ω (cid:19) (185)and G can be represented in the form G = ( v u ) (cid:18) ω ω (cid:19) ( v u ) − (186)where u = 14 (cid:18) − η (cid:19) (187)Auxiliary Lemma 7 has a completely straightforward proof, which is not pre-sented in the current work.The next two paragraphs, which are directly connected to the consistencyanalysis of the ”Restricted Middle” system, provide the additional explana-tions for subdivision into subcases ”CASE 0” , ”CASE 1” , ”CASE 2.1” and ”CASE 2.2” . According to Conclusion 3, the ”Restricted Middle” system ofequations is known to be consistent in ”CASE 0” , therefore it is sufficient toanalyse the consistency in ”CASE 1” and ”CASE 2” . Some special relations the for coefficients of conventional weight functionc(v).
It appears, that the coefficients of the conventional weight function c ( v )satisfy some special relations, which involve constant χ in ”CASE 1” andeigenvalues of the matrix G in ”CASE 2” . These relations are very useful forthe consistency analysis of the ”Restricted Middle” system. Auxiliary Lemma 8:
Let conventional weight function c ( v ) be defined by globalin-plane parametrisation ˜Π ( bicubic ) . Then for an edge with one boundary vertex,coefficients of the weight function satisfy the following relations ”CASE 1” (cid:88) k =0 χ k (cid:16) (cid:17) c k = 0 (188) ”CASE 2.1” (cid:88) k =0 (cid:0) −√ ω , (cid:1) k (cid:16) k (cid:17) c k = 0 (189) Proofs ”CASE 2.2” (cid:88) k =0 (cid:0) −√ ω (cid:1) k (cid:16) k (cid:17) c k = 0 (190) (cid:88) k =0 (cid:0) −√ ω (cid:1) k k (cid:16) k (cid:17) c k = 0 (191) Proof
See Appendix, after proof of Theorem 8 and Theorem 9.
Necessary and sufficient conditions for the consistency of the ”RestrictedMiddle” system of equations in terms of b s . The purpose of the paragraphis to formulate the necessary and sufficient conditions for the consistency of the ”Restricted Middle” system of equations in terms of the right-side coefficients ofthe full system of the indexed equations (see Equations 168 and 169). The nextthree paragraphs describe the relations between these conditions and the actualdegrees of the weight functions and finally explain the rules of classification ofthe control point ˜ C n (cid:48) − .Auxiliary Lemma 9 allows to rewrite a zero linear combination of the right-side coefficients of the ”Restricted Middle” system in terms of the right-sidecoefficients of the full system of the indexed equations. Auxiliary Lemma 9:
Let global in-plane parametrisation ˜Π ( bicubic ) be consideredand for an edge with one boundary vertex, let all non-middle control pointsbe classified and equations ”sumC-equation” , ”Eq(s)” for s = 0 , , n (cid:48) + 1 and ”Eq(n’+2)” in case of n ≥ { α s } n (cid:48) s =2 be coefficients of a non-trivial zero linear combination of rowsof the ”Restricted Middle” system of equations (cid:80) n (cid:48) s =2 α s ˆ A s = 0. Then thecorresponding linear combination of the right-side coefficients of the ”RestrictedMiddle” system of equations has the following representation in terms of theright-side coefficients of the full system of the indexed equations ”CASE 1” n (cid:48) (cid:88) s =2 α s ˆ b s = α n +2 (cid:88) s =0 χ s b s (192)where by the definition α = χ − α (193) ”CASE 2” n (cid:48) (cid:88) s =2 α s ˆ b s = ( α α ) (cid:100) n (cid:48) / (cid:101) (cid:88) s =0 G s (cid:18) b s b s +1 (cid:19) (194) Proofs where by the definition( α α ) = ( α α ) G − (195)and b n (cid:48) +2 = b = 0 in case of n = 4. Proof
See Appendix, after proof of Theorem 8 and Theorem 9.Auxiliary Lemma 10 provides an elegant necessary and sufficient conditionsfor the consistency of the ”Restricted Middle” system of equations in terms ofthe right-side coefficients of the full system of the indexed equations.
Auxiliary Lemma 10:
Let global in-plane parametrisation ˜Π ( bicubic ) be consideredand for an edge with one boundary vertex, let all non-middle control pointsbe classified and equations ”sumC-equation” , ”Eq(s)” for s = 0 , , n (cid:48) + 1 and ”Eq(n’+2)” in case of n ≥ ”Restricted Middle” system of equations is consistent if and onlyif the following conditions hold ”CASE 1” n +2 (cid:88) s =0 χ s b s = 0 (196) ”CASE 2.1” n +2 (cid:88) s =0 ( −√ ω , ) s b s = 0 (197) ”CASE 2.2” n +2 (cid:88) s =0 ( −√ ω ) s b s = 0 (198) n +2 (cid:88) s =0 ( −√ ω ) s − s b s = 0 (199) Proof
See Appendix, after proof of Theorem 8 and Theorem 9.
Relations between linear combinations of b s and coefficients of the conven-tional weight function c ( v ) . In the current paragraph, the linear combinationsof the right-sides coefficients, described in Auxiliary Lemma 10, are expandedusing the explicit formula for b s (see Equations 169 and 131). Straightfor-ward Auxiliary Lemma 11 describes relation between the linear combinationsand coefficients of the weight function c ( v ) with respect to the power basis.The Auxiliary Lemma finally justifies the fine subdivision into subcases indexedby letters (a) and (b) (see Definition 25) and makes the first step towardsexplanation of the classification rule for the control point ˜ C n (cid:48) − (see Lemma23). Proofs Auxiliary Lemma 11:
For a scalar value φ the following relations hold n +2 (cid:88) s =0 φ s b s = ” sumC − equation ” (cid:104) − ( − φ ) n +3 φ (cid:105) + φ (cid:20) n − (cid:80) i =0 φ i (cid:16) n-1i (cid:17) ∆ C i (cid:21) (cid:34) (cid:80) j =0 φ j (cid:16) (cid:17) c j (cid:35) if φ (cid:54) = − sumC − equation ”[ − ( n + 3)]+ (cid:20) n − (cid:80) i =0 ( − i i (cid:16) n-1i (cid:17) ∆ C i (cid:21) c ( power )4 +” C − equation ” (cid:104) c ( power )4 + c ( power )3 (cid:105) if φ = − n +2 (cid:88) s =0 sφ s − b s = ” sumC − equation ” (cid:104) ( − φ ) n +2 ( n +3) − ( − φ ) n +3 ( n +2)(1+ φ ) (cid:105) + φ ) (cid:20) n − (cid:80) i =0 (cid:16) φφ i − (cid:17) φ i (cid:16) n-1i (cid:17) ∆ C i (cid:21) (cid:34) (cid:80) j =0 φ j (cid:16) (cid:17) c j (cid:35) + φ (cid:20) n − (cid:80) i =0 φ i (cid:16) n-1i (cid:17) ∆ C i (cid:21) (cid:34) (cid:80) j =0 φ j j (cid:16) (cid:17) c j (cid:35) if φ (cid:54) = − sumC − equation ” (cid:2) ( n + 2)( n + 3) (cid:3) − (cid:20) n − (cid:80) i =0 ( − i ( i − i (cid:16) n-1i (cid:17) ∆ C i (cid:21) (cid:104) c ( power )4 (cid:105) − (cid:20) n − (cid:80) i =0 ( − i i (cid:16) n-1i (cid:17) ∆ C i (cid:21) (cid:104) c ( power )4 + c ( power )3 (cid:105) − ” C − equation ” (cid:104) c ( power )4 +3 c ( power )3 + c ( power )2 (cid:105) if φ = − φ = χ, ω , , ω and if one uses the specialrelations between the coefficients of the weight function c ( v ) given in AuxiliaryLemma 8 and assumes that ”sumC-equation” is satisfied. Further simplificationof Auxiliary Lemma 11 is based on some properties of the actual degrees of theweight functions, described in the next paragraph. Actual degrees of the weight functions.
It appears that the division into therefined cases given in Definition 25 is closely connected to the actual degreesof conventional weight functions.
Auxiliary Lemma 12:
Let the conventional weight functions l ( v ), r ( v ) be definedby the global in-plane parametrisation ˜Π ( bicubic ) . Then for an edge with oneboundary vertex, the maximal actual degree of the weight functions obeys thefollowing equality: max deg ( l, r ) = CASE 0 ” , ” CASE 1 . a ” , ” CASE 2 . . a ” , ” CASE 2 . . a ”1 in ” CASE 1 . b ” , ” CASE 2 . . b ”0 in ” CASE 2 . . b ” (202) Proofs Proof
See Appendix, after proof of Theorem 8 and Theorem 9.Auxiliary Lemma 12 together with Lemma 16 lead to the next Conclusion.
Conclusion 4:
Let the conventional weight function c ( v ) be defined by the globalin-plane parametrisation ˜Π ( bicubic ) . Then for an edge with one boundary vertex,the actual degree of the weight function c ( v ) satisfies the following inequalities deg ( c ) ≤ CASE 1 . b ” , ” CASE 2 . . b ”2 in ” CASE 2 . . b ” (203) Necessary and sufficient conditions for the consistency of the ”Middle” sys-tem of equations in terms of ”C-equation” and coefficients of conventionalweight function c ( v ) with respect to the power basis. Auxiliary Lemmas 10,11, 8, 12 and Conclusion 4 lead to a simple form of necessary and sufficientconditions for the consistency of the ”Middle” system.
Auxiliary Lemma 13:
Consider the in-plane parametrisation ˜Π ( bicubic ) and for anedge with one boundary vertex, let all non-middle control points be classifiedand equations ”Eq(s)” for s = 0 , , n (cid:48) + 1 and ”Eq(n’+2)” in case of n ≥ ”Middle” system of equations is consistent if and only if: • ” C − equation ” c ( power )4 = 0 (204)in the case when max deg ( l, r ) = 2 • ” C − equation ” c ( power )3 = 0 (205)in the case when max deg ( l, r ) = 1 • ” C − equation ” c ( power )2 = 0 (206)in the case when max deg ( l, r ) = 0 Proof
See Appendix, after proof of Theorem 8 and Theorem 9.Results of Auxiliary Lemma 13 finally allow to complete the proof of The-orem 8 and Theorem 9.
Proof of Theorem 8
The paragraph presents proof of both statements ofTheorem 8. (1)
According to Lemma 20, it is sufficient to show that ”sumC-equation” issatisfied in the case when deg ( c ) = 4. From the fact that deg ( l ) , deg ( r ) ≤ max deg ( l, r ) = 2 and deg ( c ) − max deg ( l, r ) = 2. The assumption of Theorem 8 implies that ”C-equation” issatisfied. According to Lemma 20, it means that ”sumC-equation” is satisfied. (2) The second statement of Theorem 8 immediately follows from AuxiliaryLemma 13. Indeed, let max deg ( l, r ) = 2. In this case Equation 204 provides Proofs a necessary and sufficient condition for the consistency of the ”Middle” system.Equation 204 is satisfied if and only if either deg ( c ) ≤ deg ( c ) = 4 and ”C-equation” is satisfied. In other words, the satisfaction of ”C-equation” inthe case when deg ( c ) − max deg ( l, r ) = 2 is a necessary and sufficient conditionfor the satisfaction of Equation 204 and so for the consistency of the ”Middle” system of equations. Similar analysis of cases when max deg ( l, r ) = 1 and when max deg ( l, r ) = 0 completes the proof of the second statement of Theorem 8. Proof of Theorem 9
The rank analysis presented in Theorem 9 directlyfollows from Conclusion 3 and from the fact that classification of the controlpoints ˜ C t ( t = 3 , . . . , n (cid:48) −
2) according to Lemma 23 guarantees the consistencyof the ”Restricted Middle” system. Like in the case of global bilinear in-planeparametrisation ˜Π ( bilinear ) , the correctness of classification of the control points( L t , R t ) ( t = 2 , . . . , n (cid:48) −
2) into basic and dependent ones can be easily verifiedby application of the Gauss elimination process to the matrix of the ”RestrictedMiddle” system. (cid:116)(cid:117)
Theorem 8 and Theorem 9
Proof of Auxiliary Lemma 5
Let α , . . . , α n (cid:48) be coefficients of a non-trivial linear combination of rows (cid:80) n (cid:48) s =2 α s ˆ A s . The linear combination is equal to zero if and only if the coefficientof every one of ∆ L s , ∆ R s ( s = 2 , . . . , n (cid:48) −
2) is equal to zero. In other words, itmeans that the linear combination is equal to zero if and only if the followingsystem of equations is satisfied(∆ L s ) α s l + 2 α s +1 l + α s +2 l = 0(∆ R s ) α s r + 2 α s +1 r + α s +2 r = 0 (207)where s = 2 , . . . , n (cid:48) − (1) or (2) of the Auxiliary Lemma is necessary for the satisfaction of the lastsystem of equations. Using the fact that l , r (cid:54) = 0 ( l , r (cid:54) = 0), one can express α s ( α s +2 ) independently of the equation corresponding to ∆ L s and the equationcorresponding to ∆ R s (see Equation 207). Equality of the expressions for α s ( α s +2 ) leads to the conclusion that the following relations should be satisfiedfor s = 2 , . . . , n (cid:48) − α s +1 g (01) + α s +2 g (02) = 0 (208) α s g (02) + 2 α s +1 g (12) = 0 (209)The following two cases are possible If g (02) (cid:54) = 0 then • g (01) (cid:54) = 0, g (12) (cid:54) = 0.On the contrary, let g (01) = 0. Then according to Equation 208, α s = 0 for s = 4 , . . . , n (cid:48) . In addition, from formulas for coefficients Proofs of ∆ L and ∆ L it follows that α = 0 and α = 0. It meansthat equality g (01) = 0 leads to a trivial linear combination whichcontradicts the assumption of Auxiliary Lemma 5. In a similarmanner it can be shown that g (12) (cid:54) = 0. • (cid:8) g (02) (cid:9) = 4 g (01) g (12) .The equality immediately follows from plugging in s = 2 into Equa-tion 208 and s = 3 into Equation 209 for s = 3. • α s +1 = χα s for s = 2 , . . . , n (cid:48) −
1, as follows from considering Equation208 for s = n (cid:48) − s = 2 , . . . , n (cid:48) − condition (1) of Auxiliary Lemma 5 is satisfied. If g (02) = 0 then • g (01) = 0, g (12) = 0.On the contrary, let g (01) (cid:54) = 0. Then according to Equation 208, α s = 0 for s = 3 , . . . , n (cid:48) −
1. From formulas for coefficients of ∆ L and ∆ L n (cid:48) − it follows that α = 0 and α n (cid:48) = 0. It means that onegets a trivial linear combination, which contradicts the assumptionof Auxiliary Lemma 5. In a similar manner it can be shown that g (12) = 0. • α s +2 = − ξα s − ηα s +1 , as it immediately follows from definitions of ξ , η and Equation 207. (Equality g (01) = g (12) = g (02) = 0 impliesthat coefficients of ∆ L s , ∆ R s given in Equation 207 are describedby the proportional expressions.)It means that condition (2) of Auxiliary Lemma 5 is satisfied.Sufficiency of condition (1) in ”CASE 1” and condition (2) in ”CASE 2” is evident since the recursive formulas for coefficients of the linear combination(see Equations 178 and 179) clearly implies that Equation 207 is satisfied. (cid:116)(cid:117) Auxiliary Lemma 5
Proof of Auxiliary Lemma 8”CASE 1”
The proof is based on representation of coefficients of the weight function c ( v )given in Equation 56. This representation implies that (cid:88) k =0 χ k c k (cid:16) k (cid:17) = (cid:104) (cid:88) i =0 χ i ˜ λ i (cid:16) i (cid:17) , (cid:88) j =0 χ j ˜ ρ j (cid:16) j (cid:17) (cid:105) (210)In order to prove that Equation 188 holds, it is sufficient to show that twovectors that participate in the last expression are parallel. According to Equa-tion 207, coefficients of the weight functions r ( v ) and l ( v ) satisfy the followingrelations (cid:80) i =0 r i χ i (cid:16) i (cid:17) = 0 , (cid:80) j =0 l j χ j (cid:16) j (cid:17) = 0 (211) Proofs Using formulas r i = −(cid:104) ˜ λ i , ˜ γ (cid:48) − ˜ γ (cid:105) and l j = (cid:104) ˜ ρ j , ˜ γ (cid:48) − ˜ γ (cid:105) , i, j = 0 , , −(cid:104) (cid:80) i =0 χ i ˜ λ i (cid:16) i (cid:17) , ˜ γ (cid:48) − ˜ γ (cid:105) (cid:104) (cid:80) j =0 χ j ˜ ρ j (cid:16) j (cid:17) , ˜ γ (cid:48) − ˜ γ (cid:105) (212)The last couple of equations means that vectors (cid:80) i =0 χ i ˜ λ i (cid:16) i (cid:17) and (cid:80) j =0 χ j ˜ ρ j (cid:16) j (cid:17) are parallel to ˜ γ (cid:48) − ˜ γ and to each other. ”CASE 2” It will be shown first that the following two equations hold ηc − ξc + 2 ξ c − ξ ηc = 0 (213)1 ξ ( ξ − η ) c + 4 1 ξ ηc − c + 4 ηc + ( ξ − η ) c = 0 (214)Like in the previous case, the proof uses expressions for coefficients of the weightfunction c ( v ) in terms of the partial derivatives of in-plane parametrisations forthe left and the right elements (see Equation 56).Let α i be the angle between ˜ γ (cid:48) − ˜ γ and − ˜ λ j and β j be the angle between ˜ ρ j and ˜ γ (cid:48) − ˜ γ (both angles are measured in the counter clockwise direction). Then l j = (cid:104) ˜ ρ j , ˜ γ (cid:48) − ˜ γ (cid:105) = | ˜ ρ j || ˜ γ (cid:48) − ˜ γ | sinβ j r i = −(cid:104) ˜ γ (cid:48) − ˜ γ, − ˜ λ i > = −| ˜ λ i || ˜ γ (cid:48) − ˜ γ | sinα i (cid:104) ˜ λ i , ˜ ρ j (cid:105) = | ˜ λ i || ˜ ρ j | sin ( α i + β j ) (215)Proof of Equation 213 makes use of two different representations of (cid:104) ˜ λ , ˜ ρ (cid:105) .First, (cid:104) ˜ λ , ˜ ρ (cid:105) may be represented as follows (cid:104) ˜ λ , ˜ ρ (cid:105) = | ˜ λ || ˜ ρ | ( sinα cosβ + cosα sinβ ) = χ (10) (cid:16) | ˜ λ || ˜ ρ | sinα cosβ + | ˜ λ || ˜ ρ | cosα sinβ (cid:17) = χ (10) (cid:16) (cid:104) ˜ λ , ˜ ρ (cid:105) + (cid:104) ˜ λ , ˜ ρ (cid:105)−| ˜ λ || ˜ ρ | cosα sinβ −| ˜ λ || ˜ ρ | sinα cosβ (cid:17) = χ (10) (cid:16) c − χ (10) (cid:104) ˜ λ , ˜ ρ (cid:105) (cid:17) = χ (10) (2 c − χ (10) c ) (216)In the same manner, it can be shown that (cid:104) ˜ λ , ˜ ρ (cid:105) = χ (12) (2 c − χ (12) c ) (217)Equations 217 and 217 imply that χ (10) (2 c − χ (10) c ) = χ (12) (2 c − χ (12) c ) (218)Substitution of χ (10) = ηξ and χ (12) = η in the last equation completes the proofof Equation 213. Proofs Equation 214 is proven in a similar manner. Coefficient c of the weight func-tion c ( v ) has the following representation (see Equation 56) c = 16 (cid:16) (cid:104) ˜ λ , ˜ ρ (cid:105) + 4 (cid:104) ˜ λ , ˜ ρ (cid:105) + (cid:104) ˜ λ , ˜ ρ (cid:105) (cid:17) Here (cid:104) ˜ λ , ˜ ρ (cid:105) + (cid:104) ˜ λ , ˜ ρ (cid:105) = | ˜ λ || ˜ ρ | ( sinα cosβ + cosα sinβ )+ | ˜ λ || ˜ ρ | ( sinα cosβ + cosα sinβ ) = χ (02) (cid:104) ˜ λ , ˜ ρ (cid:105) + χ (20) (cid:104) ˜ λ , ˜ ρ (cid:105) = χ (02) c + χ (20) c (219)4 (cid:104) ˜ λ , ˜ ρ (cid:105) = 2 χ (10) (2 c − χ (10) c ) + 2 χ (12) (2 c − χ (12) c ) (220)It means that c = (cid:104)(cid:16) χ (20) − (cid:2) χ (10) (cid:3) (cid:17) c +4 χ (10) c + 4 χ (12) c + (cid:16) χ (02) − (cid:2) χ (12) (cid:3) (cid:17) c (cid:105) (221)Substitution of χ (20) = ξ , χ (10) = ηξ , χ (02) = ξ , χ (12) = η in the last formulacompletes the proof of Equation 214.Now it is easy to prove that Equations 189, 190 and 191 are satisfied. ”CASE 2.1” Note, that ω ω = ξ (cid:54) = 0 and so in order to prove that Equation 189 is satisfiedfor ω and ω , it is sufficient to prove the equality1 ω (cid:88) k =0 ( −√ ω ) k (cid:16) k (cid:17) c k ± ω (cid:88) k =0 ( −√ ω ) k (cid:16) k (cid:17) c k = 0 (222)Using the formulas for √ ω , (see Equation 176), one can rewrite two summandsof Equation 222 in the following form ω (cid:80) k =0 (cid:0) −√ ω (cid:1) k (cid:16) k (cid:17) c k + ω (cid:80) k =0 (cid:0) −√ ω (cid:1) k (cid:16) k (cid:17) c k = − (cid:110) ξ ( ξ − η ) c + 4 ξ ηc − c + 4 ηc + ( ξ − η ) c (cid:111) (223)and ω (cid:80) k =0 (cid:0) −√ ω (cid:1) k (cid:16) k (cid:17) c k − ω (cid:80) k =0 (cid:0) −√ ω (cid:1) k (cid:16) k (cid:17) c k = − ξ (cid:112) η − ξ (cid:8) ηc − ξc + 2 ξ c − ξ ηc (cid:9) (224)The first expression is proportional to the expression given in Equation 214 andthe second one is proportional to the expression given in Equation 213, whichimplies that both of them are equal to zero. ”CASE 2.2” Proofs In this case ω = η = ξ (cid:54) = 0, √ ω = η and Equations 190 and 191 have thefollowing representations in terms of ηc − ηc + 6 η c − η c + η c = 04 η (cid:8) − c + 3 ηc − η c + η c (cid:9) = 0 (225)Using the equality η = ξ , it is easy to verify that the first expression is pro-portional to Equation 214 and the second one is proportional to η { Equation214 } + η { Equation 213 } , which implies that both of the expressions are equalto zero. (cid:116)(cid:117) Auxiliary Lemma 8
Proof of Auxiliary Lemma 9
Proof of Auxiliary Lemma 9 is quite straightforward, therefore only thebasic steps of the proof are presented below. ”CASE 1”
From the assumption that ”Eq(0)” , ”Eq(1)” , ”Eq( n (cid:48) + 1 )” and ”Eq( n (cid:48) + 2 )” (in the case of n ≥
5) being satisfied, definition of χ (see Equation 172) andformulas for ˆ b , ˆ b , ˆ bn (cid:48) − b n (cid:48) (see Equation 171), it follows that χ ˆ b + χ ˆ b = b + χb + χ b + χ b (226)ˆ b n (cid:48) − + χ ˆ b n (cid:48) = b n (cid:48) − + χb n (cid:48) + χ b n (cid:48) +1 + χ b n (cid:48) +2 (227)For the coefficients α s ( s = 2 , . . . , n (cid:48) ) of non-trivial zero linear combination of therows of the ”Restricted Middle” system, the corresponding linear combinationof the right-side coefficients can be written as (cid:80) n (cid:48) s =2 α s ˆ b s = α (cid:80) n (cid:48) s =2 χ s ˆ b s = α (cid:110)(cid:80) n (cid:48) +1 s =0 χ s b s + χ n (cid:48) +2 b n (cid:48) +2 (cid:111) = α (cid:80) n +2 s =0 χ s b s (228)Equation 228 is based on the fact that ˆ b s = b s for s = 4 , . . . , n (cid:48) − α s +1 = χα s for s = 2 , . . . , n (cid:48) − ”CASE 2” Similarly to the previous case, the main difficulty in the proof is connected tothe expression of the first and the last couples of ˆ b s in terms of b s .The assumption that equations ”Eq(s)” for s = 0 , , n (cid:48) + 1 and for s = n (cid:48) + 2in case of n ≥ b = b + η − ξξ b − ηξ b ˆ b = b + ηξ b − ξ b ˆ b n (cid:48) − = b n (cid:48) − − ξb n (cid:48) +1 + 2 ξηb n (cid:48) +2 ˆ b n (cid:48) = b n (cid:48) − ηb n (cid:48) +1 + (4 η − ξ ) b n (cid:48) +2 (229) Proofs The first couple of relations clearly implies that( α α ) (cid:18) ˆ b ˆ b (cid:19) = ( α α ) (cid:20)(cid:18) b b (cid:19) + G (cid:18) b b (cid:19)(cid:21) (230)If n (cid:48) is odd then the last couple of relations (see Equation 229) can be rewrittenin the form (cid:18) ˆ b n (cid:48) − ˆ b n (cid:48) (cid:19) = (cid:18) b n (cid:48) − b n (cid:48) (cid:19) + G (cid:18) b n (cid:48) +1 b n (cid:48) +2 (cid:19) (231)If n (cid:48) is even then it can be shown that( α n (cid:48) − α n (cid:48) − ) (cid:18) ˆ b n (cid:48) − ˆ b n (cid:48) − (cid:19) + α n (cid:48) ˆ β n (cid:48) =( α n (cid:48) − α n (cid:48) − ) (cid:20)(cid:18) b n (cid:48) − b n (cid:48) − (cid:19) + G (cid:18) b n (cid:48) b n (cid:48) +1 (cid:19)(cid:21) (232)Equation 232 makes use of the fact that the satisfaction of ”sumC-equation” leads to equality b n (cid:48) +2 = 0. Note, that it is the only step of the proof wherethe satisfaction of ”sumC-equation” is required. It implies that requirement ofthe satisfaction of ”sumC-equation” in the formulation of Auxiliary Lemma 9is necessary only in case of even n ≥ α s α s +1 ) = ( α α ) G s , s = 2 , . . . , n (cid:48) − (cid:116)(cid:117) Auxiliary Lemma 9
Proof of Auxiliary Lemma 10
Proof of Auxiliary Lemma 10 is based on the first statement of the algebraicAuxiliary Lemma 1, which provides the necessary and sufficient conditions forthe consistency of a system in terms of the linear combinations of rows andright-side coefficients. ”CASE 1”
Necessity and sufficiency of Equation 196 immediately follows from Equation192. ”CASE 2”
According to Auxiliary Lemma 9, it is clear that the ”Restricted Middle” systemis consistent if and only if equation (cid:100) n (cid:48) / (cid:101) (cid:88) s =0 G s (cid:18) b s b s +1 (cid:19) = (cid:18) (cid:19) (233)is satisfied. ”CASE 2.1” Proofs Proof of the Auxiliary Lemma in ”CASE 2.1” makes use of the Jordan formof matrix G given in Equation 183. Using the formula( v v ) − = 1 ξ ( √ ω − √ ω ) (cid:18) √ ω −√ ω ω −√ ω √ ω ω (cid:19) (234)one sees that Equation 233 is equivalent to the system (cid:18) (cid:19) = (cid:100) n (cid:48) / (cid:101) (cid:80) s =0 (cid:18) ω s ω s (cid:19) (cid:18) √ ω −√ ω ω −√ ω √ ω ω (cid:19) (cid:18) b s b s +1 (cid:19) = (cid:100) n (cid:48) / (cid:101) (cid:80) s =0 (cid:18) √ ω (cid:0) ( −√ ω ) s b s + ( −√ ω ) s +1 b s +1 (cid:1) −√ ω (cid:0) ( −√ ω ) s b s + ( −√ ω ) s +1 b s +1 (cid:1) (cid:19) (235)The last system in turn is clearly equivalent to the following equation, whichshould be satisfied for ω and ω system of equations (cid:100) n (cid:48) / (cid:101) +1 (cid:88) s =0 (cid:0) −√ ω , (cid:1) s b s = 0 (236)It remains to show, that the upper bound of summation can be changed from2 (cid:100) n (cid:48) / (cid:101) +1 to n +2. The substitution of the upper bound is valid because b = 0for n = 4 and b n +2 = 0 for an even n > ”sumC-equation” is satisfied). ”CASE 2.2” Proof of the Auxiliary Lemma in ”CASE 2.2” is very similar to the proof in ”CASE 2.1” . In the current case the Jordan form of the matrix G is given byEquation 186. Using the formula( v u ) − = 12 √ ω (cid:18) √ ω −√ ω ω (cid:19) (237)one sees that Equation 233 is equivalent to the system (cid:18) (cid:19) = (cid:100) n (cid:48) / (cid:101) (cid:80) s =0 (cid:18) ω s sω s − ω s (cid:19) (cid:18) √ ω −√ ω ω (cid:19) (cid:18) b s b s +1 (cid:19) = (cid:32) √ ω (cid:80) (cid:100) n (cid:48) / (cid:101) +1 s =0 ( −√ ω ) s b s − (cid:80) (cid:100) n (cid:48) / (cid:101) +1 s =0 ( −√ ω ) s − sb s −√ ω (cid:80) (cid:100) n (cid:48) / (cid:101) +1 s =0 ( −√ ω ) s b s (cid:33) (238)Replacement of the upper bound of summation by n + 2 (which is valid due tothe same reasons as in ”CASE 2.1” ) allows to conclude that Equations 198and 199 provide the necessary and sufficient conditions for the consistency ofthe ”Restricted Middle” system. (cid:116)(cid:117) Auxiliary Lemma 10
Proof of Auxiliary Lemma 12
Auxiliary Lemma 12 makes use of the following pair of implications
Proofs ”Impl(1)” max deg ( l, r ) ≤ ”CASE 1” or ”CASE 2” are satisfied. ”Impl(2)” max deg ( l, r ) = 0 implies that conditions of case ”CASE 2.2.b” are satisfied.A short proof of ”Impl(1)” and ”Impl(2)” is given below. ”Impl(1)” Condition max deg ( l, r ) ≤ l ( power )2 = l − l + l = 0, r ( power )2 = r − r + r = 0. Therefore r l ( power )2 = l r ( power )2 and equality g (01) = g (02) holds. In the same manner multiplication of l ( power )2 and r ( power )2 by r and l respectively leads to equality g (12) = g (02) . It means that the condition max deg ( l, r ) ≤ g (01) = g (12) = 12 g (02) (239)Equation 239 holds if either g (01) = g (12) = g (02) = 0 which corresponds to ”CASE 2” or g (01) , g (12) , g (02) simultaneously differ from zero and (cid:0) g (02) (cid:1) =4 g (01) g (12) which corresponds to ”CASE 1” . ”Impl(2)” Condition max deg ( l, r ) = 0 means that l ( power )2 = r ( power )2 = l ( power )1 = r ( power )1 = 0 and so l = l = l , r = r = r (240)It implies that g (01) = g (12) = g (02) = 0 and that ξ = χ (02) = 1, η = χ (12) = 1,which corresponds to ”CASE 2.2.b” The remaining part of the proof presents computations of the exact value of max deg ( l, r ) in every one of the possible cases. ”CASE 0” In this case max deg ( l, r ) = 2 as it immediately follows from ”Impl(1)” . ”CASE 1” According to ”Impl(2)” , max deg ( l, r ) ≥
1. It remains to showthat max deg ( l, r ) (cid:54) = 1 in ”CASE 1.a” and max deg ( l, r ) (cid:54) = 2 in ”CASE1.b” . ”CASE 1.a” On the contrary, let max deg ( l, r ) = 1. As it follows fromthe proof of ”Impl(1)” , in this case g (01) = g (12) = g (02) . Conditions of ”CASE 1” imply that g (01) , g (12) , g (02) (cid:54) = 0. It means that constant χ is correctly defined and by the definition is equal to χ = − g (01) g (02) = − ”CASE 1.a” . ”CASE 1.b” On the contrary, let max deg ( l, r ) = 2. From the condi-tions of ”CASE 1.b” , it follows that 2 g (01) = g (02) , 2 g (12) = g (02) . Ad- Proofs dition of l r and l r to the both parts of the first and second equalitiesleads to the conclusion that l r ( power )2 = r l ( power )2 , l r ( power )2 = r l ( power )2 (241)The last couple of equations means that both l ( power )2 and r ( power )2 arenot equal to zero (otherwise from Equation 241 it follows that l ( power )2 = r ( power )2 = 0 which contradicts the assumption that max deg ( l, r ) = 2)and that l r = l r = l ( power )2 r ( power )2 (242)Equation 242 implies that g (02) = 0 which contradicts the conditions of ”CASE 1” . ”CASE 2””CASE 2.1” Let conditions of ”CASE 2.1” be satisfied. From theformula √ ω , = η ± (cid:112) η − ξ (see Equation 176) it follows that conditionsof subcase ”CASE 2.1.a” are equivalent to inequality1 − η + ξ (cid:54) = 0 (243)and conditions of subcase ”CASE 2.1.b” are equivalent the equality1 − η + ξ = 0 (244)Using Equations 243, 244 and the explicit formulas for ξ and η (seeEquation 174), one sees that conditions of subcase ”CASE 2.1.a” aresatisfied if and only if the following inequality holds (in addition to thegeneral conditions of ”CASE 2.1” ) l ( power )2 l = r ( power )2 r (cid:54) = 0 (245)and conditions of subcase ”CASE 2.1.b” are satisfied if and only if thefollowing equality holds (in addition to the general conditions of ”CASE2.1” ) l ( power )2 l = r ( power )2 r = 0 (246)Now it is easy to compute the value of max deg ( l, r ) for both subcases of ”CASE 2.1” . ”CASE 2.1.a” Equation 245 clearly implies that max deg ( l, r ) = 2. ”CASE 2.1.b” Equation 246 implies that max deg ( l, r ) ≤
1. Ac-cording to ”Impl(2)” , max deg ( l, r ) (cid:54) = 0. It means that max deg ( l, r ) =1. Proofs ”CASE 2.2””CASE 2.2.a” On the contrary, let max deg ( l, r ) ≤
1. It impliesin particular that0 = l ( power )2 l = l − l + l l = ξ − η + 1 = η − η + 1 = ( η − (247)The last equality means that √ ω = η = 1, which contradicts theconditions of subcase ”CASE 2.2.a” . ”CASE 2.2.b” In this case √ ω = η = 1 and so l = l , r = r . Inaddition, ξ = η = 1 and so l = l , r = r . It clearly implies that max deg ( l, r ) = 0. (cid:116)(cid:117) Auxiliary Lemma 12
Proof of Auxiliary Lemma 13
The ”Middle” system of equations is composed of the ”Restricted Middle” system and ”sumC-equation” . It means that the consistency of the ”Middle” system is equivalent to the couple of the following conditions • ”sumC-equation” is satisfied. • The necessary and sufficient conditions for the consistency of the ”Re-stricted Middle” system, given in Auxiliary Lemma 10, hold.Now Auxiliary Lemma 13 can be proven by a simple analysis of all possiblesubcases. Below the proof in ”CASE 2.1.a” is given; the rest of the subcasescan be analysed in a similar manner. ”CASE 2.1.a”
According to Auxiliary Lemma 12, max deg ( l, r ) = 2. Aux-iliary Lemmas 10 and 11 imply that if ”sumC-equation” is satisfied then the ”Restricted Middle” system of equations is consistent if and only if0 = (cid:80) n +2 s =0 ( −√ ω , ) s b s = ” sumC − equation ” (cid:104) − ( √ ω , ) n +3 −√ ω , (cid:105) + −√ ω , (cid:104)(cid:80) n − i =0 ( −√ ω , ) i (cid:16) n-1i (cid:17) ∆ C i (cid:105) (cid:104)(cid:80) j =0 ( −√ ω , ) j (cid:16) (cid:17) c j (cid:105) (248)From the relation between coefficients of the weight function c ( v ) given in Aux-iliary Lemma 8 (see Equation 189) it follows that Equation 248 holds if andonly if ”sumC-equation” is satisfied. It means that the ”Middle” system ofequations is consistent if and only if ”sumC-equation” is satisfied. Accordingto Lemma 20, one sees that the ”Middle” system is consistent if and only if” C − equation ” c ( power )4 = 0. (cid:116)(cid:117) Auxiliary Lemma 13
Proof of the correctness of Algorithm 6
Proofs ”Stage 1” According to Lemma 21, at every inner vertex, local templates forclassification of V , E , D , T -type control points remains unchanged with respectto the case of global bilinear in-plane parametrisation ˜Π ( bilinear ) . ThereforeAlgorithms from Section 12 can be reused. ”Stage 1” of Algorithm 6 includes ”Stage 1” and ”Stage 2” of algorithm forconstruction of global MDS in case of global bilinear in-plane parametrisation˜Π ( bilinear ) (Algorithm 1), applied to all inner vertices.Global classification of V , E -type control points succeeds for any n ≥
4. For n ≥
5, application of local templates for classification of D , T -types controlpoints never leads to a contradictions (see Subsection 12.1). For n = 4, globalclassification of D , T -type control points succeeds in the current case accordingto Theorem 6, because no D -relevant boundary vertices are involved. ”Stage 2” According to Lemma 22, local templates for classification of the middle control points for edges with two inner vertices remain unchanged withrespect to the case of global bilinear in-plane parametrisation ˜Π ( bilinear ) . There-fore ”Stage 2” of Algorithm 6 is precisely ”Stage 3” of Algorithm 1, appliedto all edges with two inner vertex, which is known to succeed for any n ≥ ”Stage 3” It is the only stage, which requires modifications with respect toconstruction of MDS in the case of global bilinear in-plane parametrisation˜Π ( bilinear ) . Classification of the control points at ”Stage 3” is made locally,traversal order of the edges with one boundary vertex is not important.For every edge with one boundary vertex, the correctness of classificationfollows from the fact, that after application of ”Step 1” all initial assumptionsof Lemma 23 and then (after classification of the ”central” middle controlpoints) of Theorem 9 are satisfied. (cid:116)(cid:117) Algorithm 6
Proofs (a) (b) (c)(d) (e) Fig. 54: An illustrations to proof of Theorem 6.
Computation of Thin Plate energy for bilinear in-plane parametrisation of mesh element E Computation of Thin Plate energy for bilinear in-planeparametrisation of mesh element
The following example demonstrates the linear form of energy functional in thecase of the Thin Plate problem (see Subsection 19.1 for the precise definition)and presents explicit computational formulas for the case of bilinear in-planeparametrisation.The current Section analyses a single mesh element and therefore it is pos-sible to use slightly different notations than in the rest of the work - (
X, Y, Z )instead of ( P X , P Y , P ), J ( u, v ) instead of J ( ˜ P ) ( u, v ) - in order to make formulasmore compact and clear. In addition, the superscript of Bernstein polynomialswill sometimes be omitted.Let ˜Π be a fixed global regular parametrisation, ¯Ψ ∈ ¯ FUN ( n ) ( ˜Π) and ¯ P =( ˜ P , Z ) = ¯Ψ | ˜ p be the restriction of ¯Ψ on some mesh element ˜ p ∈ ˜ Q . In-planeparametrisation of the element ˜ P = ( X, Y ) = ˜Π | ˜ p is fixed.Energy functional has the following form in terms of Z -components of thecontrol points E = n (cid:88) i,j =0 n (cid:88) k,l =0 Z i,j Z k,l a klij − n (cid:88) i,j Z ij b ij . (249)where a klij = D (cid:82) (cid:82) ˜ p (cid:16) ∂ B ij ∂X + ∂ B ij ∂Y (cid:17) (cid:16) ∂ B kl ∂X + ∂ B kl ∂Y (cid:17) − (1 − ν ) (cid:16) ∂ B ij ∂X ∂ B kl ∂Y + ∂ B ij ∂Y ∂ B kl ∂X − ∂ B ij ∂X∂Y ∂ B kl ∂X∂Y (cid:17) dXdYb ij = (cid:82) (cid:82) ˜ p f B ij dXdY. (250)In matrix form one gets: E = Z T AZ − Z T B (251)where Z = ( Z , . . . , Z nn ) T - vector of Z -components of all the control pointsof the patch; symmetric ( n + 1) × ( n + 1) square matrix A is built fromcomponents a klij and vector B = ( b , . . . , b nn ) T .Elements of matrix A and vector B can be computed by representing theintegrals with respect to variables ( u, v ) and then by applying any appropriatenumerical method. Lemma 27 contains some useful general formulas for partialderivatives of u , v and Bernstein polynomials with respect to X and Y . Lemma 27:
Let ˜ P ( u, v ) = ( X ( u, v ) , Y ( u, v )) be a regular, twice differentiableparametrisation of a planar mesh element and J ( u, v ) = (cid:18) ∂X∂u ∂Y∂u∂X∂v ∂Y∂v (cid:19) (252)is Jacobian corresponding to the parametrisation. Then Computation of Thin Plate energy for bilinear in-plane parametrisation of mesh element (1) The first-order and the second-order partial derivatives of u and v withrespect to X and Y can be computed according to the following formulas (cid:18) ∂u∂X ∂v∂X∂u∂Y ∂v∂Y (cid:19) = J − ( u, v ) = det ( J ( u,v )) (cid:18) ∂Y∂v − ∂Y∂u − ∂X∂v ∂X∂u (cid:19)(cid:32) ∂ u∂X ∂ v∂X ∂ u∂X∂Y ∂ v∂X∂Y (cid:33) = ∂J − ( u,v ) ∂X = − J − ( ∂J∂u ∂u∂X + ∂J∂v ∂v∂X ) J − (cid:32) ∂ u∂X∂Y ∂ v∂X∂Y∂ u∂Y ∂ v∂Y (cid:33) = ∂J − ( u,v ) ∂Y = − J − ( ∂J∂u ∂u∂Y + ∂J∂v ∂v∂Y ) J − (253) (2) The first-order and the second-order partial derivatives of Bernstein poly-nomial B ( u, v ) can be computed according to the following formulas ∂B∂X = ∂B∂u ∂u∂X + ∂B∂v ∂v∂X∂B∂Y = ∂B∂u ∂u∂Y + ∂B∂v ∂v∂Y∂ B∂X = ∂ B∂u (cid:0) ∂u∂X (cid:1) + 2 ∂ B∂u∂v ∂u∂X ∂v∂X + ∂ B∂v (cid:0) ∂v∂X (cid:1) + ∂B∂u ∂ u∂X + ∂B∂v ∂ v∂X ∂ B∂X∂Y = ∂ B∂u ∂u∂X ∂u∂Y +2 ∂ B∂u∂v (cid:0) ∂u∂X ∂v∂Y + ∂u∂Y ∂v∂X (cid:1) + ∂ B∂v ∂v∂X ∂v∂Y + ∂B∂u ∂ u∂X∂Y + ∂B∂v ∂ v∂X∂Y∂ B∂Y = ∂ B∂u (cid:0) ∂u∂Y (cid:1) + 2 ∂ B∂u∂v ∂u∂Y ∂v∂Y + ∂ B∂v (cid:0) ∂v∂Y (cid:1) + ∂B∂u ∂ u∂Y + ∂B∂v ∂ v∂Y (254)In the case of bilinear in-plane parametrisation ˜Π ( bilinear ) (see Section 9) thefollowing formulas take place. For a planar quadrilateral element with vertices˜ A, ˜ B, ˜ C, ˜ D (see Figure 4), determinant of Jacobian det ( J ( u, v )) may be writtenas det ( J ( u, v )) = (cid:104) ˜ B − ˜ A, ˜ C − ˜ D (cid:105) u + (cid:104) ˜ C − ˜ B, ˜ D − ˜ A (cid:105) v + (cid:104) ˜ B − ˜ A, ˜ D − ˜ A (cid:105) (255) det ( J ( u, v )) is a linear function in terms of u and v in a general case and constantin the case of a parallelogram. It is important that det ( J ( u, v )) is a linear and nota bilinear function, because det ( J ( u, v )) participates in expressions for partialderivatives and its order is essential for the choice of an appropriate methodfor numerical or exact integration when the coefficients of energy matrix arecomputed.Based on Equation 255, Lemma 28 presents the explicit formulas for thefirst-order and the second-order partial derivatives of u and v with respect to X and Y . Lemma 28:
Let a convex planar quadrilateral element have bilinear in-planeparametrisation ˜ P ( u, v ) = ( X ( u, v ) , Y ( u, v )), J ( u, v ) be Jacobian correspondingto the parametrisation and ˜ T = ˜ t ( ˜ A, ˜ B, ˜ C, ˜ D ) = ˜ A − ˜ B + ˜ C − ˜ D be the twistcharacteristic of the element (see Subsection 2.3.1). Then the first-order andthe second-order partial derivatives of u and v with respect to X and Y can becomputed according to the following formulas (cid:18) ∂u∂X ∂v∂X∂u∂Y ∂v∂Y (cid:19) = 1 det ( J ( u, v )) (cid:16) T Y u + ( D Y − A Y ) − T Y v − ( B Y − A Y ) − T X u − ( D X − A X ) T X v + ( B X − A X ) (cid:17) (256) ∂ u∂X ∂ v∂X ∂ u∂X∂Y ∂ v∂X∂Y∂ u∂Y ∂ v∂Y = − ∂u∂X ∂v∂X∂u∂X ∂v∂Y + ∂u∂Y ∂v∂X ∂u∂Y ∂v∂Y ( T X , T Y ) (cid:18) ∂u∂X ∂v∂X∂u∂Y ∂v∂Y (cid:19) (257) Computation of Thin Plate energy for bilinear in-plane parametrisation of mesh element Technical Lemma 28 allows concluding that the first-order partial derivativesare the linear rational functions and the second-order partial derivatives arecubic rational functions, where the denominator changes more rapidly when thequadrilateral is less similar to a parallelogram.
From MDS to solution of the quadratic minimisation problem F From MDS to solution of the quadratic minimisationproblem
Let Z all denote Z -components of all the control points of all the patches, orderedso that Z all = ( Z (1)00 , . . . , Z (1) nn , . . . , Z ( p max )00 , . . . , Z ( p max ) nn ) T (258)where Z ( p ) ij denotes a control point belonging to the patch with order number p = 1 , . . . , p max . Here no dependencies between control points of the differentpatches are assumed. Let A ( p ) and B ( p ) ( p = 1 , . . . , p max ) be respectively thematrix and the vector which correspond to the computation of the energy func-tional for the patch with order number p . Then the global energy functionalcan be written in the form E ( Z all ) = p max (cid:88) p =1 E ( p ) = Z Tall AZ all − Z Tall B (259)where A = A (1) . . . A (2) . . . . . . A ( p max ) , B = B (1) B (2) ... B ( p max ) (260)Formal differentiating of the energy functional and setting the result equal tozero yields ∂ E ∂ Z all = 2 AZ all − B = 0 (261)Let ˜ B ( n ) be a minimal determining set which fits a chosen ”additional” con-straints. The set of Z -coordinates corresponding to in-plane control points fromthe MDS will be denoted by Z basis . From the algebraic point of view, Z basis de-scribes degrees of freedom of the constrained minimisation problem, where only G continuity constraints are applied. Dependencies of the remaining controlpoints on Z basis has a linear form and it is possible to define the dependencymatrix C so that Z all = CZ basis (262)It leads to the energy functional of the form E ( Z basis ) = Z Tbasis C T ACZ basis − Z
Tbasis C T B (263)Differentiation of the functional gives ∂ E ∂ Z basis = 2 C T ACZ basis − C T B = 0 (264) From MDS to solution of the quadratic minimisation problem Note, that dependency of a dependent control point on the basic control pointsis usually defined ”step by step”. At every stage of the classification process,for a control point which gets dependent status it is sufficient to define howit explicitly depends on the basic control points and control points which aredefined as dependent during the previous stages or steps of the classification. Itclearly leads to the gradual construction of the final dependency matrix. Z basis is known to fit the considered ”additional” constraints. Therefore, itonly remains to separate Z basis into two subsets Z fixed and Z free = Z basis \Z fixed , where Z fixed is Z -coordinates of the control points which should befixed as a result of application of the ”additional” constraints. Assuming that Z basis is ordered so that Z basis = (cid:18) Z free Z fixed (cid:19) and C = ( C free C fixed ) (265)one gets the final linear system of equations with unknowns Z free ∂ E ∂ Z free = 2 C Tfree AC free Z free + 2 C Tfree AC fixed Z fixed − C Tfree B = 0 (266)Here A is n all × n all symmetric square matrix, B is n all × C free is n all × n free matrix and C fixed is n all × n fixed matrix, where n all , n free , n fixed are numbers of variables in Z all , Z free , Z fixed respectively. Construction of an interpolating surface: current approach and interpolation based on D mesh of curves G Construction of an interpolating surface: current approachand interpolation based on D mesh of curves As already mentioned above, from the pure theoretical point of view results ofthe current work fit the general theory presented in Part II and have a niceand simple geometrical interpretation.The current Section concentrates on the more practical aspects of construc-tion of the resulting surface for the (vertex)(tangent plane)-interpolation prob-lem. The (vertex)(tangent plane)-interpolation problem is chosen since it is the”native” problem for the approach based on interpolation of 3 D mesh of curvesand one of the possible applications of our method. The Section highlightsthe similarities and the differences between the implementation of the currentapproach and the interpolation method described in work [35]. Algorithm for construction of a smooth interpolant presented in work [35]
In the case of a cubic mesh of curves, work [35] provides an algorithm for theconstruction of G -smooth quadratic interpolant. The algorithm requires thatthe mesh of curves satisfies the following requirements.- The mesh curves define a unique tangent plane at every mesh vertex.- Sufficient vertex enclosure constraint (see 2) holds at every even meshvertex.- At every mesh vertex, tangents of any two sequential curves emanatingfrom the vertex span an angle of less than π .For a quadrilateral mesh, the algorithm defines (3 , , , , ”TangentsRelation” (cid:104) ¯ (cid:15) ( L ) , ¯ (cid:15) ( R ) (cid:105)(cid:104) ¯ (cid:15) ( R ) , ¯ (cid:15) ( C ) (cid:105) + (cid:104) ¯ (cid:15) ( C ) , ¯ (cid:15) ( L ) (cid:105) = (cid:104) ¯ (cid:15) (cid:48) ( L ) , ¯ (cid:15) (cid:48) ( R ) (cid:105)(cid:104) ¯ (cid:15) (cid:48) ( R ) , ¯ (cid:15) (cid:48) ( C ) (cid:105) + (cid:104) ¯ (cid:15) (cid:48) ( C ) , ¯ (cid:15) (cid:48) ( L ) (cid:105) (267)holds (see Figure 55). Here the tangent vectors at two endpoints of the curveare considered in coordinates of the tangent plane they belong to.Generally, the coefficients of the weight functions are fixed in accordance withthe given mesh data. The B´ezier control points are built locally and linearly,proceeding from the boundary control points to the interior of a patch. Anyunder-constraint situation is solved by some local heuristic, like averaging or theleast-square technique. In some situations, one additional heuristic constrainthas to be imposed : the choice of the twist control points adjacent to an eveninner mesh vertex; the choice of two middle ”side” control points adjacent toa mesh curve which satisfies the ”Tangents Relation” ; the choice of the innermiddle control point of a quartic B´ezier patch. Comparison with the current approach
Although both techniques solve the(vertex)(tangent plane)-interpolation problem by construction of a G -smoothpiecewise B´ezier surface, they pursue different goals and are applicable in dif-ferent situations . The approach presented in work [35] serves mainly for the Construction of an interpolating surface: current approach and interpolation based on D mesh of curves construction of nicely looking smooth surfaces, it may be used as a tool of visu-alization. The current approach finds the constrained solution of some minimi-sation problem defined by a given energy functional. There are some similaritiesand many differences in these two techniques . Local/global nature of the solution. Application of the local heuristics.
In-terpolation of 3 D mesh of curves with a smooth surface uses local techniques.Control points of the resulting B´ezier patches are constructed separately forevery face of the mesh and depend on the geometry of the mesh. The solutionof all the under-constraint situations and the choice of the inner control pointsis based on some local heuristics. The current approach uses no heuristics, butrequires solution of some global system. Construction of a MDS allows to de-crease significantly the number of variables and to solve the global system interms of the basic control points only. Under-constraint situations and additional degrees of freedom
There are def-inite similarities between under-constraint situations of the mesh interpolationalgorithm and some special cases studied in the current work. For example,an additional basic twist control point for an inner even vertex (in the currentapproach, see Subsection 11.2.3) reflects the application of a local heuristic forthe choice of the twist control points adjacent to an even inner vertex (in the3 D mesh of curves interpolation approach). An additional middle basic controlpoint for an edge, which satisfies the ”Projections Relation” (see Theorem 5)is the application of a local heuristic for the choice of the ”side” middle controlpoints adjacent to a mesh curve which satisfies the ”Tangents Relation” . Requirements on the initial data.
The approach based on interpolation of3 D mesh of curves works only if the mesh is admissible; the satisfaction of thevertex enclosure constraint, which involves the first and second-order derivatives,should be verified at every vertex of the mesh. The current approach doesnot define any requirements on the initial data (excepting minor natural meshlimitations). Moreover curves which participate in 3 D mesh of curves are fullydefined; in particular it means that the first and second-order derivatives ofthe resulting patches in the boundary directions are initially defined. In thecurrent approach, a boundary curve of a resulting patch has at least one degreeof freedom (at least one inner control point of the curve is not fixed by the initialdata), which allows to always satisfy the vertex enclosure constraint. Degrees of the resulting patches.
According to the current approach, there ex-ist an instance of ˜ B (4) ( ˜Π ( bilinear ) ) (in the case of a mesh with a polygonal globalboundary) and an instance of ˜ B (5) ( ˜Π ( bicubic ) ) or mixed MDS ˜ B (4 , ( ˜Π ( bicubic ) ) (incase of a mesh with a smooth global boundary), which fit the (vertex)(tangent-plane)-interpolation condition. The approach for interpolation of 3 D mesh ofcurves allows to construct the solution of degree 4 for any admissible mesh; acase of a mesh with a given smooth global boundary is not an exception.The main reason for the difference is that the approaches use different tech-niques for definition of the weight functions. As soon as a 3 D mesh of curvesappears to be admissible, the approach of curve mesh interpolation defines co-efficients of the weight functions which participate in the vertex enclosure con-straints separately at every mesh vertex and then proceeds with definition ofthe weight functions independently for every curve. In the current approach,the weight functions of the different edges of the same mesh element are inter- Construction of an interpolating surface: current approach and interpolation based on D mesh of curves connected by an in-plane parametrisation of the element. In the case of a planarmesh with a piecewise-cubic boundary, the bicubic in-plane parametrisation ofthe boundary mesh elements leads to the high degrees of the weight functionsfor the edges with one boundary vertex. Of course, for every such edge it mightbe possible to construct the weight functions separately, precisely in the samemanner as in the case of global bilinear in-plane parametrisation. For example,in Figure 56, the weight functions of edge (˜ γ, ˜ γ (cid:48) ) are defined by the bilinearparametrisations of virtual mesh elements constructed according to the middlecontrol points of the boundary curve. Although these weight functions and cor-respondent G -continuity equations could be analysed similarly to the analysispresented in Section 4, they lead to a disagreement of in-plane control pointscorresponding to the weight functions of three inner edges of the boundary meshelement. It implies that in-plane parametrisation is no longer correctly definedfor boundary mesh elements, while the definition of in-plane parametrisation apriory plays the principal role for the current approach. Construction of an interpolating surface: current approach and interpolation based on D mesh of curves Fig. 55: An illustration for the ”Tangents Relation””Tangents Relation”