Curvature for everyone
Arizona State University
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.