Sfb 288 Differential Geometry and Quantum Physics |
Geodesics and waves
on piecewise linear surfaces
For the numerical integration of a vector field v on a curved surface with e.g. an Euler Method geodesics may be used in the basic step as follows:
An Example: Discrete Geodesics on Zoll's Surfaces
Zoll's surfaces [O.Zoll,1903] have the property that all their geodesics are simply closed and have same length. They come in a family of rotational surfaces whose meridians are given as solutions of an ODE. It is remarkable that the family contains surfaces which have locally strict negative gaussian curvature. The closedness property of geodesics on a Zoll's suface makes them suitable candidates for testing the concept of discrete geodesics.
Author: Konrad Polthier, Markus Schmies