Ulrich Reif

A unified approach to subdivision algorithms near extraordinary vertices

We present a unified approach to subdivision algorithms for meshes with arbitrary topology which admits a rigorous analysis of the generated surface and give a sufficient condition for the regularity of the surface, i.e. for the existence of a regular smooth parametrization near the extraordinary point. The criterion is easily applicable to all known algorithms such as those of Doo-Sabin and Catmull-Clark, but will also be useful to construct new algorithms like interpolatory subdivision schemes.

The simplest subdivision scheme for smoothing polyhedra

Given a polyhedron, construct a new polyhedron by connecting every edge-midpoint to its four neighboring edge-midpoints. This refinement rule yields a <italic>C</italic><supscrpt>1</supscrpt> surface and the surface has a piecewise quadratic parametrozation except at a finite number of isolated points. We analyze and improve the construction.

Weighted Extended B-Spline Approximation of Dirichlet Problems

We describe a new finite element method which uses weighted extended B-splines on a regular grid as basis functions for solving Dirichlet problems on bounded domains in arbitrary dimensions. This web-method does not require any grid generation and can be implemented very efficiently. It yields smooth, high order accurate approximations with relatively low dimensional subspaces.

Analysis of Algorithms Generalizing B-Spline Subdivision

A new set of tools for verifying smoothness of surfaces generated by stationary subdivision algorithms is presented. The main challenge here is the verification of injectivity of the characteristic map. The tools are sufficiently versatile and easy to wield to allow, as an application, a full analysis of algorithms generalizing biquadratic and bicubic B-spline subdivision. In the case of generalized biquadratic subdivision the analysis yields a hitherto unknown sharp bound strictly less than 1 on the second largest eigenvalue of any smoothly converging subdivision.

Shape characterization of subdivision surfaces: case studies

For subdivision surfaces, it is important to characterize local shape near flat spots and points where the surface is not twice continuously differentiable. Applying general principles derived in [Computer Aided Geometric Design, doi:10.1016/j.cagd.2004.04.006, in press], this paper characterizes shape near such points for the subdivision schemes devised by Catmull and Clark and by Loop. For generic input data, both schemes fail to converge to the hyperbolic or elliptic limit shape suggested by the geometry of the input mesh: the limit shape is a function of the valence of the extraordinary point rather than the mesh geometry. We characterize the meshes for which the schemes behave as expected and indicate modifications of the schemes that prevent convergence to the wrong shape. We also introduce a type of chart that, for a specific scheme, can help a designer to detect early when a mesh will lead to undesirable curvature behavior.

Best bounds on the approximation of polynomials and splines by their control structure

Abstract We present best bounds on the deviation between univariate polynomials, tensor product polynomials, Bezier triangles, univariate splines, and tensor product splines and the corresponding control polygons and nets. Both pointwise estimates and bounds on the Lp-norm are given in terms of the maximum of second differences of the control points. The given estimates are sharp for control points corresponding to arbitrary quadratic polynomials in the univariate case, and to special quadratic polynomials in the bivariate case.

Polynomial degree reduction in the L 2 -norm equals best Euclidean approximation of Bézier coefficients

Abstract Given a polynomial p of degree n we want to find a best L 2 -approximation over the unit interval from polynomials of degree m . This problem is shown to be equivalent to the problem of finding the best Euclidean approximation of the vector of Bernstein–Bezier coefficients of p from the vector of degree-raised Bernstein–Bezier coefficients of polynomials of degree m .

Biquadratic G-spline surfaces

Abstract We present a simple procedure for generating geometrically smooth surfaces based on biquadratic rectangular Bezier patches from control meshes of arbitrary topological type. Geometrical smoothness conditions are used only near the irregular vertices of the mesh, elsewhere the standard tensor product setup is sufficient.

Nonuniform web-splines

The construction of weighted extended B-splines (web-splines), as recently introduced by the authors and J. Wipper for uniform knot sequences, is generalized to the nonuniform case. We show that web-splines form a stable basis for splines on arbitrary domains in Rm which provides optimal approximation power. Moreover, homogeneous boundary conditions, as encountered frequently in finite element applications, can be satisfied exactly by using an appropriate weight function. To illustrate the performance of the method, it is applied to a scattered data fitting problem and a finite element approximation of an elliptic boundary value problem.

A degree estimate for subdivision surfaces of higher regularity

Subdivision algorithms can be used to construct smooth surfaces from control meshes of arbitrary topological structure. In contrast to tangent plane continuity, which is well understood, very little is known about the generation of subdivision surfaces of higher regularity. This work presents a degree estimate for piecewise polynomial subdivision surfaces saying that curvature continuity is possible only if the bi-degree d of the patches satisfies d > 2k+2, where k is the order of smoothness on the regular part of the surface. This result applies to any stationary or non-stationary scheme consisting of masks of arbitrary size provided that some generic symmetry and regularity assumptions are fulfilled.