Basic ideas from the theory of differential geometry will crop up repeatedly in the material we’ll be covering this semester. In order to better prepare you to understand our upcoming papers, we’ll be spending a day on a quick review of some basic differential geometry material. Keep in mind: this is a large area of mathematics, and we’re only covering the most basic stuff.
Our discussion for today will focus primarily on the classical theory for continuous surfaces. We’ll also begin discussing — and this discussion will continue throughout the semester — how to go about discretizing these ideas on polygonal meshes.
The following paper by Meyer et al. provides a nice introduction to a discretization of classical differential geometry to meshes.
There are any number of books available on this topic, and they all have more or less the same title. I’ve put together some of my own notes that try to summarize the most basic material that you would encounter in these books. It also contains a brief section with thoughts on which books you might try reading.
For a nice discussion on adapting the usual notions of vector calculus (e.g., div/grad/curl) to manifolds, I recommend reading: