Tamás Várady

A Multi-sided Bézier Patch with a Simple Control Structure

A new n‐sided surface scheme is presented, that generalizes tensor product Bézier patches. Boundaries and corresponding cross‐derivatives are specified as conventional Bézier surfaces of arbitrary degrees. The surface is defined over a convex polygonal domain; local coordinates are computed from generalized barycentric coordinates; control points are multiplied by weighted, biparametric Bernstein functions. A method for interpolating a middle point is also presented.

Enhancement of a multi-sided Bézier surface representation

Abstract A new multi-sided surfacing scheme – the Generalized Bezier (GB) patch – has been introduced recently by Varady et al. (2016) . The patch is created over a polygonal domain, its parameterization is defined by means of generalized barycentric coordinates. It has a simple control structure; the control points are associated with a combination of bi-parametric Bernstein functions, multiplied by rational terms. GB patches are compatible with adjacent quadrilateral Bezier patches and inherit most of their properties. In this paper we present an enhanced version of the former scheme. The control structure has been slightly modified, yielding a perfect generalization of quadrilateral patches. The parameterization has also been altered, matching a concept of how quadrilateral domains transform to n-sided polygons. We propose improved blending functions and investigate how the weight deficiency of the basis functions can be distributed amongst control points. The former rational weighting functions have been modified to support not only G 1 , but higher degree continuity between adjacent patches. After briefly discussing how degree reduction and elevation proceed, we present algorithms to automatically create and optimize the internal control points of GB patches. A few simple examples and suggestions for future work conclude the paper.

Ribbon-based transfinite surfaces ☆

Abstract One major issue in CAGD is to model complex objects using free-form surfaces of general topology. A natural approach is curvenet-based design, where designers directly create and modify feature curves. These are interpolated by smoothly connected, multi-sided patches, which can be represented by transfinite surfaces, defined as a combination of side interpolants or ribbons. A ribbon embeds Hermite data, i.e., prescribed positional and cross-derivative functions along boundary curves. The paper focuses on two transfinite schemes: the first is an enhanced and extended variant of a multi-sided generalization of the classical Coons patch ( Varady et al., 2011 ); the second one is based on a new concept of combining doubly curved composite ribbons, each one interpolating three adjacent sides. Main contributions include various ribbon parameterizations that surpass former methods in quality and computational efficiency. It is proven that these surfaces smoothly interpolate the prescribed ribbon data. Both formulations are based on non-regular convex polygonal domains and distance-based, rational blending functions. A few examples illustrate the results.

A general framework for constrained mesh parameterization

Parameterizing or flattening a triangle mesh is necessary for many applications in computer graphics and geometry. While mesh parameterization is a very popular research topic, the vast majority of the literature is focused on minimizing distortion or satisfying constraints related to certain applications such as texturing or quadrilateral remeshing. Certain downstream applications require adherence to more general, geometric constraints -- possibly at the cost of higher distortion. These geometric constraints include requirements such as certain vertices lie on some line or circle, or a planar curve or developable region keeps its shape during parameterization. We present a framework for enforcing such constraints, motivated by the As-Rigid-As-Possible parameterization method, and demonstrate its effectiveness through several examples.

G2 Surface Interpolation Over General Topology Curve Networks

The basic idea of curve network‐based design is to construct smoothly connected surface patches, that interpolate boundaries and cross‐derivatives extracted from the curve network. While the majority of applications demands only tangent plane (G1) continuity between the adjacent patches, curvature continuous connections (G2) may also be required. Examples include special curve network configurations with supplemented internal edges, “master‐slave” curvature constraints, and general topology surface approximations over meshes.

Transfinite surface interpolation with interior control

Graphical abstractDisplay Omitted Highlights? Transfinite surface patches are used for interpolating 3D free-form curve networks. ? New distance-based blending functions make possible to edit the patch interior. ? Auxiliary points and curves provide localized shape control. ? Auxiliary surfaces are added while boundary constraints are retained. ? One- and two-sided surfaces are created using special parameterizations. There are various techniques to design complex free-form shapes with general topology. In contrast to the approaches based on trimmed surfaces and control polyhedra, in curve network-based design feature curves can be directly created and edited in 3D. Multi-sided patches interpolate this curve network with slopes given by associated tangent ribbons. The patches are smoothly connected and yield a natural and predictable surface model. This paper focuses on special design techniques to adjust the interior of transfinite patches when further shape control is needed. While the boundary constraints are retained, additional vertices, curves and even interior control surfaces are supplemented to gain more design freedom. The main idea is to apply different distance-based blending functions with special parameterizations over non-regular, n-sided domains. This concept can be naturally extended to create one- and two-sided patches as well. Shape variations will be demonstrated by a few simple examples.

P-curves and surfaces: Parametric design with global fullness control

Abstract A new curve representation (P-curves) that is well-suited for computer-aided geometric design is proposed. While several properties of Bezier and B-spline curves are inherited, new useful features have also been introduced. A P-curve is defined by a control polygon; it forms a single C ∞ continuous segment with endpoint interpolation. The new basis functions have been inspired by the Mean Value generalized barycentric coordinates. P-curves actually represent a family of curves with a continuously changing fullness parameter that determines the proximity between the curve and its control polygon. It is fairly straightforward to increase the degree of design freedom of P-curves, as the new control point will always be inserted on a selected chord retaining both the full control polygon and the shape of the curve. In this paper, we describe the construction of P-curves and prove their basic mathematical properties. Several examples will be shown to compare P-curves with Bezier and B-spline curves. The adjustment of the fullness parameter will also be demonstrated. The new basis functions can also be used to define tensor product P-surfaces with a global control to loosely or tightly approximate the control grid.

P-Bézier and P-Bspline curves – new representations with proximity control

Abstract Proximity curves represent a family of curves that are associated with a given parametric curve, defined by control points and basis functions. Proximity curves continuously sweep from this curve to its control polygon depending on a proximity value, that determines the location of an intermediate curve and its fullness (or tension). The proximity value also determines sensitivity, i.e. how strongly the shape is affected by displacing control points. An important feature of proximity curves relates to the insertion of new control points: while in other schemes a new degree of freedom leads to repositioning the existing control points, in our case the new control points are always placed on some chord of the control polygon. Our first proximity curve scheme – called P-curves – has been published recently ( Kovacs and Varady, 2017 ), having C ∞ continuity and G 1 endpoint interpolation. The basis functions were constructed by means of generalized barycentric coordinates, and a somewhat limited algorithm for control point insertion was proposed. Our current paper takes a different approach: the basis functions are calculated by a much simpler algebra that is capable to reproduce standard formulations like Bezier and B-splines curves, and can maintain C n end constraints. We introduce the Proximity-Bezier, shortly P-Bezier, and P-Bspline curves, including the construction of basis functions and the most important mathematical properties. A general control point insertion algorithm is also described. Several examples are shown to compare the classical and the new representations. A tensor product generalization of the scheme is also demonstrated.

Parameterizing and extending trimmed regions for tensor-product surface fitting

Abstract Our goal is to approximate the data points of an irregular, trimmed triangular mesh by standard, tensor-product surfaces, such as NURBS. This is a difficult and ambiguous task driven by several parameters, including tolerances, knot vectors, smoothing terms, etc. One of the most crucial issues is the parameterization of the data points that will have a strong influence on the quality of the surface to be fitted. We propose new techniques that attempt to establish some correspondence between the given trimmed region and the unknown four-sided surface. In the first phase a four-sided virtual guiding frame is created in 3D using labeled boundary segments. Then a 2D parameterization is computed that minimizes distortion while satisfying constraints implicitly defined by the labels and the frame. In the second phase new data points are inserted to fill in the domain rectangle, and an inverse mapping to 3D is performed. This leads to a four-sided mesh that extends the original trimmed region in a smooth and natural manner, and is well-suited for fitting an untrimmed tensor-product surface. We will discuss several examples to illustrate the benefits of our techniques using a “black-box” surface fitter. Benefits include even curvature distribution, natural surface extensions in the vicinity of the trimming boundaries, avoiding wiggles for the full surface and a tight bounding box.