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Based on a kind of Q-Bézier surfaces with shape parameters, the basic properties of the surfaces and the geometric significance of the shape parameters are analyzed. To resolve the problem of shape control and adjustment of composite surfaces, the continuity conditions for Q-Bézier surfaces of degree (m, n) are investigated. Taking advantage of the terminal properties of generalized Bernstein basis functions, we derive the conditions of G1 and G2 continuity between two adjacent Q-Bézier surfaces. In addition, the specific steps of smooth continuity between Q-Bézier surfaces and the shape adjustment function of shape parameters for composite surfaces are discussed. The modeling examples show that the proposed smooth continuity conditions are not only intuitive and easy to implement, but also greatly enhance the shape adjustability, which provide a useful method for constructing complex surfaces in engineering design. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to show how to use the Gauss map to produce efficient triangulations of parametric surfaces. The central idea is to invert canonical triangulations of the sphere using the Gauss map. The main problem is that the inverse of the Gauss map is not in general well defined. We show how to overcome this problem and formulate an algorithm to generate the desired triangulation. [ABSTRACT FROM PUBLISHER]

Abstract: When bivariate filter banks and wavelets are used for surface multiresolution processing, it is required that the decomposition and reconstruction algorithms for regular vertices derived from them have high symmetry. This symmetry requirement makes it possible to design the corresponding multiresolution algorithms for extraordinary vertices. Recently lifting-scheme based biorthogonal bivariate wavelets with high symmetry have been constructed for surface multiresolution processing. If biorthogonal wavelets have certain smoothness, then the analysis or synthesis scaling function or both have big supports in general. In particular, when the synthesis low-pass filter is a commonly used scheme such as Loop’s scheme or Catmull–Clark’s scheme, the corresponding analysis low-pass filter has a big support and the corresponding analysis scaling function and wavelets have poor smoothness. Big supports of scaling functions, or in other words big templates of multiresolution algorithms, are undesirable for surface processing. On the other hand, a frame provides flexibility for the construction of “basis” systems. This paper concerns the construction of wavelet (or affine) bi-frames with high symmetry. In this paper we study the construction of wavelet bi-frames with 4-fold symmetry for quadrilateral surface multiresolution processing, with both the dyadic and refinements considered. The constructed bi-frames have 4 framelets (or frame generators) for the dyadic refinement, and 2 framelets for the refinement. Namely, with either the dyadic or refinement, a frame system constructed in this paper has only one more generator than a wavelet system. The constructed bi-frames have better smoothness and smaller supports than biorthogonal wavelets. Furthermore, all the frame algorithms considered in this paper are given by templates so that one can easily implement them. [Copyright &y& Elsevier]

We discuss the problem of fitting a curve or surface to given measurement data. In many situations, the usual least-squares approach (minimization of the sum of squared norms of residual vectors) is not suitable, as it implicitly assumes a Gaussian distribution of the measurement errors. In those cases, it is more appropriate to minimize other functions (which we will call norm-like functions) of the residual vectors. This is well understood in the case of scalar residuals, where the technique of iteratively re-weighted least-squares, which originated in statistics (Huber in Robust statistics, 1981) is known to be a Gauss–Newton-type method for minimizing a sum of norm-like functions of the residuals. We extend this result to the case of vector-valued residuals. It is shown that simply treating the norms of the vector-valued residuals as scalar ones does not work. In order to illustrate the difference we provide a geometric interpretation of the iterative minimization procedures as evolution processes. [ABSTRACT FROM AUTHOR]

Abstract: Rationally parameterized hypersurfaces can be classified with respect to their RC properties (Rational Convolutions) with the help of the Gröbner bases theory. This classification focuses on special classes of rational parameterizations which provide a rational description of convolution hypersurfaces generally (GRC parameterizations), or just in some special cases (SRC parameterizations). The main aim of this paper is to bring the theory of the so-called PN surfaces (surfaces with Pythagorean Normal vectors) and their PN parameterizations (parameterizations fulfilling the PN condition) in relation to the theory of SRC parameterizations and to show that this type of parameterizations can be further classified with respect to the degree of the construction of convolution surfaces. The connection of SRC PN parameterizations to the well-known concepts of proper and square-root parameterizations is also investigated. [Copyright &y& Elsevier]

Point and splat-based representations have become a suitable technique both for modeling and rendering complex 3D shapes. Converting other kinds of models as parametric surfaces to splat-based representations will allow to mix surface and splat-based models and to take advantage of the existing point-based rendering methods. In this work, we present an approach to convert a parametric surface into a splat-based representation. It works in parametric space, performs an adaptive sampling based on the surface curvature and a given error tolerance and uses power Voronoi diagrams. The goal is to approximate the surface with an optimized set of elliptical splats. [ABSTRACT FROM AUTHOR]

In this paper two curvature adaptive methods of surface triangulation are presented. Both methods are based on edge refinement to obtain a triangulation compatible with the curvature requirements. The first method applies an incremental and constrained Delaunay triangulation and uses curvature bounds to determine if an edge of the triangulation is admissible. The second method uses this function also in the edge refinement process, i.e. in the computation of the location of a refining point, and in the re-triangulation needed after the insertion of this refining point. Results are presented, comparing both approaches. [ABSTRACT FROM AUTHOR]

This paper develops a robust dual representation for the tangent space of a rational surface. This dual representation of tangent space is a very useful tool for visibility analysis. Visibility constructs that are directly derivable from the dual representation of this paper include silhouettes, bitangent developables and kernels. It is known that the tangent space of a surface can be represented by a surface in dual space, which we call a tangential surface. Unfortunately, a tangential surface is usually infinite. Therefore, for robust computation, the points at infinity must be clipped from a tangential surface. This clipping requires two complementary refinements, the first to allow clipping and the second to do the clipping. First, three cooperating tangential surfaces are used to model the entire tangent space robustly, each defined within a box. Second, the points at infinity on each tangential surface are clipped away while preserving everything that lies within the box. This clipping only involves subdivision along isoparametric curves, a considerably simpler process than exact trimming to the box. The isoparametric values for this clipping are computed as local extrema from an analysis using Sederberg’s piecewise algebraic curves. A construction of the tangential surface of a parametric surface is outlined, and it is shown how the tangential surface of a Bézier surface can be expressed as a rational Bézier surface. [ABSTRACT FROM AUTHOR]

We present a simple, accurate and efficient algorithm for collision detection among moving ellipsoids. Its efficiency is attributed to two results: (i) a simple algebraic test for the separation of two ellipsoids, and (ii) an efficient method for constructing a separating plane between two disjoint ellipsoids. Inter-frame coherence is exploited by using the separating plane to reduce collision detection to simpler subproblems of testing for collision between the plane and each of the ellipsoids. Compared with previous algorithms (such as the GJK method) which employ polygonal approximation of ellipsoids, our algorithm demonstrates comparable computing speed and much higher accuracy. [ABSTRACT FROM AUTHOR]

We present a biorthogonal wavelet construction for Loop subdivision, based on the lifting scheme. Our wavelet transform uses scaling functions that are recursively defined by Loop subdivision for arbitrary manifold triangle meshes. We orthogonalize our wavelets with respect to local scaling functions. This way, the wavelet analysis computes locally a least squares fit when reducing the resolution and converting geometric detail into sparse wavelet coefficients. The contribution of our approach is local computation of both, wavelet analysis and synthesis in linear time. We provide numerical examples supporting the stability of our scheme. [ABSTRACT FROM AUTHOR]

Surface flattening is a crucial problem for many applications as indicated by the steady flow of new methods appearing in related publications. Quality control of these methods, by means of “accuracy criteria” independent of particular flattening methodologies, has not been addressed yet by researchers. This is exactly the subject of this paper: a detailed analysis of flattening is presented leading to geometric and physics-based criteria. These are implemented in intuitively-acceptable visualization techniques, which are applied to practical examples. [ABSTRACT FROM AUTHOR]

We present an algorithm for computing the convex hull of freeform rational surfaces. The convex hull problem is reformulated as one of finding the zero-sets of polynomial equations; using these zero-sets we characterize developable surface patches and planar patches that belong to the boundary of the convex hull. [ABSTRACT FROM AUTHOR]

An algorithm for constructing arbitrary parametric quadratic quadric triangles in rational Bézier form is presented. The algorithm does not require the knowledge of the underlying quadric, an important property in view of applying this method for the interpolation of triangulated 3D data points. The algorithm consists of four steps starting with the arbitrary choice of the three corner points and corner weights of the patch, by then constructing a certain triangle and a tetrahedron by means of which the remaining inner control points and weights are obtained guaranteeing the resulting patch to lie on a quadric surface. [ABSTRACT FROM AUTHOR]