Discrete Curve Flows in Two-Dimensional Cayley–Klein Geometries

Abstract

Using the method of equivariant moving frames, we study geometric flows of discrete curves in the nine Cayley–Klein planes. We show that, under a certain arc-length preserving flow, the curvature invariant κ _n evolves according to the differential-difference equation $\frac {\partial \kappa _n}{\partial t} = (1+\epsilon \kappa _{n+1}^2)(\kappa _{n+1}-\kappa _{n-1})$, where the value of 𝜖 ∈{−1, 0, 1} is linked to the geometry of the Cayley–Klein plane.