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)”

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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.