Numerical computation of surface curvature

Question

In 2 dimensions, the definition of curvature of a curve $y = y(x)$ is

\begin{equation} C = \frac{y''}{(1+y'^{2})^{3/2}} \end{equation}

and it is easy to estimate the curvature numerically for given set of data points.

I need to find surface curvature numerically for some surface data points. I read on wikipedia that there are more than one definition of curvatures in 3 dimensions namely, normal curvature, geodesic curvature and geodesic torsion. Can some one please explain in simple terms how different these definitions arise and how to determine most suitable definition for given purpose?

Answer 1

If your surface is given parametrically by explicit mathematical expressions, then the formulas given in any differential geometry book can be used. If you need to approximate derivatives, you're in the realm of numerical differentiation.

If your surface is given as a polygonal mesh, then you're in the realm of Discrete Differential Geometry, which has a different set of formulas and algorithms. See for instance:

• The Discrete Differential Geometry Forum

• Polygon Mesh Processing, especially Differential Geometry slides

• (Discrete) Differential Geometry slides