Impulse Gauss Curvatures 2002 SSHE-MA Conference

Abstract: In Riemannian (differential) geometry, the differences between Euclidean geometry, elliptic geometry, and hyperbolic geometry are understood in terms of curvature. I think Gauss and Riemann captured the essence of geometry in their studies of surfaces and manifolds, and their point of view is spectacularly illuminating. Unfortunately, curvature is highly non-trivial to work with. I will talk about a more accessible version of curvature that dates back to Descartes. Curvature The Gauss curvature K is a generalization to surfaces of the curvature κ for curves that is covered in calculus. The curvature for the graph of a function f is closely related to the concavity, and since f' ' is the derivative of the slope of the tangent line, the concavity tells us how fast the slope is changing. In other words, it is a measure of how much the curve is curving. The concavity, however, tells us the rate of curvature relative to distances along the x-axis. Therefore, the relationship between concavity and the shape of the