Curvature for everyone

Abstract

If y''=0, then y is linear. But if the second derivative is not zero, then its value does not measure the distance from being linear in a geometric way: thinking graphically, just consider the asymptotically straight graphs of polynomial and exponential functions, contrasted by second derivatives that do not go to zero.

Curvature is the geometric measure of how far a smooth object is from being linear. Geometrically, curvature may be defined in very simple, most elegant ways -- but algebraic formulae are typically messy, although basically straightforward. As such curvature is predestined to be studied with the aid of a computer algebra system (CAS). Indeed, we feel that by relying on CAS, curvature may regain the more prominent place in our curricula that it used to have, and it still deserves: understanding linearity in all its guises is a key objective of our curricula -- and this also includes a solid and intuitive sense that quantifies the distance from being linear. From the antique (helices as constant curvature curves and Dido's problem), through such classics as Gauss, all the way to general relativity (curved space-time) and most recently optimal control, curvature has been, and still is a core subject of mathematics. CAS at our finger-tips now make it accessible for everyone.

In this talk we take a graphically oriented approach to curvature, leaving the lengthy technical calculations to the CAS. Starting with curvature of plane and space curves, we proceed to curvature of surfaces imbedded in 3-space. The highlight of the talk is an interactive visualization of the interaction of curvature (coded by color) with properties of the geodesic spheres and geodesic spray. The central property of interest is that positive curvature "focuses" geodesics (or light-rays), whereas negative curvature is a sufficient condition for optimality of geodesics. We end with a brief preview of how such CAS-aided visually-oriented approach nicely extends to the next steps: curvature of abstract manifolds and curvature of optimal control.