Geometric methods in computer graphics and numerical analysis

Abstract

We study some geometric problems arising in computer graphics and numerical analysis. In the first part of this thesis, we consider optimal surface triangulations and suggest a new optimal criterion for surface triangulation. The optimal triangulation will be a critical point of an energy function defined on triangulations. The energy function gives the statistical variance of edge length for a triangulation. Since the function value of a triangulation is the minimal spring energy when each edge of a triangulation is made of the same kind of spring, the optimal triangulation is a solution of a stability problem. We study optimal triangulations for surfaces in $R^3$ and its applications to numerical analysis and computer graphics. For the unit sphere, regular polyhedra with triangular faces are obtained as optimal ones, especially the soccer ball structure is studied. The optimal triangulation can be curvature adapted triangulation for arbitrary surfaces. So, our optimal triangulations are useful for computer graphics and computer aided geometric design. In addition, we generate a planar optimal grid with respect to a given Riemannian metric, and show that the optimal grid is effective when solving elliptic problems with finite element methods. In the second part, surface extension problem is considered. We suggest a new method to extend a given surface with $C^2$ continuity along the boundary. The extended part is generated by special quadratic or cubic splines satisfying the boundary conditions up to first order. Furthermore, it is proved that the normal curvature of the extended surface can be controlled by boundary conditions and a control parameter. As a practical application, we extend a surface patch representing a correction lens inside the Color Display Tube of a monitor set so that the extended surface can be mounted in a safe and easy way.