Can anyone give a quick summary of what you can do with this theory? I know a bit about geometry (up to the basics of the De Rahm cohomology and the Riemann curvature tensor, say).
What does the discrete version get you? What are its applications to CS?
> What does the discrete version get you? What are its applications to CS?
Everything! When you are programming a computer, everything must be discrete. If you need any differential geometry, it is discrete differential geometry then. You may want to hide this fact and pretend that your stuff is continuous, but at some point you will be computing derivatives by evaluating a function on nearby points. In that case, discrete differential geometry tells you which weights to put in your difference scheme.
> If you need any differential geometry, it is discrete differential geometry then
This is a bit oversimplified/exaggerated.
We need to use discrete bits in our representation for a computer, but our numbers can be the coefficients of continuous functions or relations (e.g. polynomials or trigonometric polynomials), and so it is possible to represent continuous functions to whatever precision we have compute resources to handle without “discretizing” per se.
in a past life i learned a little about PDE and the theory justifying why finite element methods work to approximate solutions.
i never explicitly ran into anything about discrete differential geometry, but that doesn't mean it wasn't there all along, lurking beneath (or perhaps above?) my level of understanding.
some reading: Evans -- Partial Differential Equations, Wendland -- Scattered Data Approximation, Wahba -- Spline models for observational data .
Applications of discrete curvature have been around a while, for example this PDF [0] from 2003, so maybe this paper provides theoretical context for existing work.
[0] "Anisotropic Polygonal Remeshing"
Pierre Alliez, David Cohen-Steiner, Olivier Devillers, Bruno Lévy, Mathieu Desbrun
it seems useful to distinguish between applying transforms to meshes, or not. There seems to be a lot of interest in using math to improve meshes for surface analysis, but there are also many problems that are not like that, hence identify up-front...
What does the discrete version get you? What are its applications to CS?