Local parametric bézier interpolants for triangular meshes: from polynomial to rational schemes

Problems of "Reverse Engineering" type are recurrent in Computer Aided (Geometric) Design (CA(G)D) and in computer graphics, in general. They consist in the reconstruction of objects from point clouds. In computer graphics, for visualisation purposes, for example, the existing solutions consist in triangulating the point data and then fitting them with planar triangles. The object is thus approximated by a piecewise linear surface, which is only C0 continuous. In order to obtain a smooth aspect a huge amount of triangles is necessary. Triangular meshes are widely used because they are sufficiently general to represent surfaces of arbitrary genus. The goal of this thesis, after having acquired an overview of the existing literature, was to present a scattered data interpolation method by means of polynomial and rational parametric surfaces in Bézier form of the lowest possible degree. Every method that tries to solve a data fitting problem encounters the same main difficulty: dealing with the smoothness of the surface. To be useful for surface design, a data fitting scheme must produce a smooth surface. After a brief introduction, in chapter 2 we analyse the existing continuous interpolatory curved shape surface schemes. They recently emerged to address specific requirements of the resource-limited hardware environments and to provide smooth surfaces by visually enhancing the resulting C0 surface by using as little information as possible. The bibliographic study allowed us also to analyse what is called vertex consistency problem. This problem is about the limitations involved when constructing G1-continuous surfaces by means of triangular Bézier patches. The G1 methods proposed until now in the literature either bypass the problem or find the way to construct the surface in such a way that it is solvable. In chapter 3, we briefly describe the interesting recently published solutions and we focus our attention on quadratic patches by analysing some particular G1-conditions and describing our first attempts to solve them. Then, in chapter 4 we treat G1 rational blend interpolatory schemes, i.e., those methods that use rational blends to construct the surface avoiding the vertex consistency problem. The study of the existing schemes allowed us to develop a new cubic polynomial Gregory patch. Its generalisation to a rational patch is currently a work in progress. The first results to improve the surface shape of our schemes on arbitrary meshes, preserving its good approximation behaviour and, possibly, keeping its computational cost as low as possible, are shown in chapter 5. Finally, in chapter 6 we conclude summarising and commenting the work presented in this thesis.