Discrete differential geometry of curves and surfaces

Abstract

Discrete differential geometry is an emerging field which seeks to study the discrete analogues of concepts arising from differential geometry, such as the curvature of a discrete curve or surface. Discrete curves can intuitively be thought of as a finite number of points in space, connected by line segments where each point is connected to a maximum of two segments. Similarly, discrete surfaces may be thought of as a set of points in space connected by line segments, where these line segments form polyhedra. The points are referred to as vertices, the line segments are referred to as edges and the polyhedra are referred to as faces. Beyond its rich mathematical theory, discrete differential geometry has various applications which range from computer graphics and geometry processing to freeform architecture. In this dissertation, we introduce the discrete differential geometry of curves and surfaces which leads to an application involving discrete minimal surfaces. Here we prove that the discrete minimal catenoid converges to the smooth minimal catenoid as the discretization is refined. We also use the main theorem in this section to generate some discrete catenoids at different levels of refinement.