Curvature CS 468 Lecture 8 Notes Scribe : Andy Nguyen

In the previous lecture we developed the notion of the curvature of a surface, showing a bunch of nice theoretical properties. But aside from theoretical elegance, what’s so great about curvature? Well, remember from the plane curve setting that the curvature function uniquely defines a curve parameterized by arc length up to a rigid motion. A similar result holds for surfaces, though the proof is beyond the scope of this course: If you know the curvature in every direction at every point on the surface, then you’ve encoded all of the geometry of the surface. Recall that the curvature of a surface at a point can be fully encoded in two scalars (maximum and minimum curvature) and two directions (principal curvature directions); this means that this small list of values encodes all the geometry in a format that is meaningful to manipulate directly. One application is using curvature as a descriptor: Recall that we derived two scalars from the maximum and minimum curvature, namely Gaussian curvature (the product) and mean curvature (the arithmetic mean). Both of these have intuitive interpretations: Gaussian curvature is positive when the surface is parabolic (convex/concave), and negative when it is hyperbolic (saddle-shaped); while mean curvature describes the extent to which a surface bends. Another application is using curvature as an alternate representation of a surface in which to perform operations such as smoothing before converting back into Euclidean space (though be warned, this back-conversion is highly non-trivial) [9]. Alternatively, we can perform smoothing directly on the surface by minimizing curvature subject to keeping the surface “close" to the original, since curvature is a measure of the deviation of the surface from “flatness", which is the extreme of smoothness [10]. While these applications thus far have focused on the curvature values, we can also find applications for the curvature directions. For one, these principal curvature directions can be used to trace out principal curves, which tend to follow primary geometric curvatures; as a result, we can use these principle curves to create highlights in stylized renderings [2]. For another, since we know that the principle curvature directions are orthogonal and lie in the tangent plane, we can use these directions as a grid of sorts on a surface, which we can use to guide remeshing, a particularly nasty problem because it involves both combinatorial changes (changing the mesh structure) and continuous changes (changing the vertex locations) [1].