Discrete Methods in Differential Geometry

Abstract

Nowadays, differential geometry is not only still one of the most profound research areas of mathematics after having had great influence in physics for more than a century, but it has also recently begun to play a very important role in computer graphics and image processing. The Poincare conjecture was believed to be proven by Perelman in 2004. However, other mathematicians are still looking into the details of the proof where not all parts are constructive. Researchers in digital topology have already started to explore the possibility of proving this conjecture in digital or discrete cases. This would also be exciting since a pure digital proof must be able to be implemented in terms of algorithms and would be constructive. In this chapter, we introduce the basic knowledge of differential geometry and some practical topics in its applications to computer graphics and computer vision. Due to the fact that differential geometry has a close relationship to variational analysis and harmonic functions, we also include a brief review of the principle of variational analysis. This chapter emphasizes some important topics of the discrete methods in differential geometry including circle packing, curvature flow, and minimum surface calculations.