Modeling With Triangular B-Splines: Theory And Implementation

In the past 35 years, many researches have been conducted on multivariatc splines in Computer Aided Geometric Design (CAGD) and Numerical Computation fields. Fruitful articles and book have been presented till now. Nowadays, splines of lower degree have been applied pervasively in both industrial field and academic field such as Computer Vision and Graphics, Digital Image Processing, Computer Animation, Computer Aided Industrial Manufacturing, Computer Geometric Modeling, Reverse Engineering, Approximation Theory, Computational Geometry and etc. CAGD has evolved from a brand of Applied Mathematics to an independent subject with multiple cross fields.Many parametric curves and surfaces have been developed in CAGD. Surfaces over triangulations have been paid special interests by many researchers for the simplex is the simplest polytope in the real affine space. Any complex polytope can be decomposed into a set of adjacent simplices and the surfaces over arbitrary topological domain are obtained via triangulation. Many surfaces over triangulations were already developed in the past, but the most basic multivariate B-Splines, Simplex Splines, have not gain much popularity for some important reasons in implementation. The Triangular B-Spline Surfaces are built upon the Normalized Simplex Spline basis and they can represent the surfaces over arbitrary topological domain. We tried to make up the gap between the theory and implementation of bivariate Triangular B-Spline Surfaces. A system of cubic bivariate Triangular B-Spline Surfaces has been developed with performance good enough for practice.The geometric nature of Triangular B-Spline Surfaces is stressed in our system for future development. In this thesis, the Triangular B-Spline Surfaces are introduced through geometric approach. Some useful propositions and properties of the theory of Triangular B-Spline Surfaces we found in the implementation arc presented and an efficient framework to model with Triangular B-Spline Surfaces is generated from the theoretical backgrounds. Meanwhile, the blossoming is given deeply discussion in this thesis.In general, we have adopted a theory-implementation approach to design our system and the efficiency of our system enables us to use Triangular B-Spline Surfaces for computer geometric modeling in the future.