Just as curvature provides powerful ways to describe and analyze smooth surfaces, discrete curvatures provide a powerful way to encode and manipulate digital geometry—and is a fundamental component of many modern algorithms for surface processing. This first of two lectures on discrete curvature from the integral viewpoint, i.e., integrating smooth expressions for discrete curvatures in order to obtain curvature formulae suitable for discrete surfaces. In the next lecture, we will see a complementary variational viewpoint, where discrete curvatures arise by instead taking derivatives of discrete geometry. Amazingly enough, these two perspectives will fit together naturally into a unified picture that connects essentially all of the standard discrete curvatures for triangle meshes.
2 thoughts on “Lecture 15—Discrete Curvature I (Integral Viewpoint)”
We probably said this in class (can’t remember) but on slide 20, discrete mean curvature and discrete Gauss curvature are swapped (discrete mean curvature is denoted by a $K$, and discrete Gauss curvature is denoted by a $H$).
Thanks Josh. I fixed this issue and a couple others in the slides above. I also added a slide at the very end that gives the Steiner formula for smooth rather than polyhedral surfaces, using the expressions we derive in the earlier slides.