Discrete Differential Geometry: An Applied Introduction

Columbia University, 2008. - 103 pages. This new and elegant area of mathematics has exciting applications, as this course demonstrates by presenting practical examples in geometry processing (surface fairing, parameterization, and remeshing) and simulation (of cloth, shells, rods, and fluids). The behavior of physical systems is typically described by a set of continuous equations using tools such as geometric mechanics and differential geometry to analyze and capture their properties. For purposes of computation one must derive discrete (in space and time) representations of the underlying equations. Researchers in a variety of areas have discovered that theories, which are discrete from the start, and have key geometric properties built into their discrete description can often more readily yield robust numerical simulations which are true to the underlying continuous systems: they exactly preserve invariants of the continuous systems in the discrete computational realm. Such theories make up the nascent field of discrete differential geometry.