Lectures in Discrete Differential Geometry 1 – Plane Curves

The classic theory of differential geometry concerns itself with smooth curves and surfaces. In practice, however, our experiments can only measure a finite amount of data, and our simulations can only resolve a finite amount of detail. Discrete differential geometry (DDG) studies discrete counterparts of classical differential geometry that are applicable in this discrete setting, and converge to the smooth theory in the limit of refinement. Smooth geometric objects possess a rich set of symmetries, invariants, and interrelationships – for instance, the Gauss-Bonnet theorem ties together the Gaussian curvature of a surface to its topology in a beautiful way. There are many different ways to discretize any given geometric object, and surprisingly, it is often possible with care to choose a discretization with special properties that exactly, not just approximately, mimic this smooth structure. Choosing the right discretization that preserves the right structure leads to particularly elegant and efficient algorithms for solving problems in computational geometry and physical simulation. I will give an overview of DDG, with a particular focus on discretizing the geometry of surfaces in R. Topics we will cover include discrete curvature measures, discrete exterior calculus, the Laplace-Beltrami operator and its properties, mean curvature flow, conformal surface parameterization, vibration modes of membranes, statics and dynamics of elastic plates, and time integration using the discrete Hamilton’s principle. Before studying discrete surfaces, however, we will look at the geometry of curves in the plane, and in this more elementary setting gain initial experience with DDG.