Sfb 288 Differential Geometry and Quantum Physics

Discrete Curves
Bäcklund Transformations for Discrete Curves

Given a discrete curve g (a polygon) and a complex number l. Then for any startingpoint t₀ one can evolve the condition that every two neigbouring points of g and the corresponding points of t have crossratio l. We call t a Bäcklund transform of g. Given a closed curve, there are in general two other closed Bäcklund transform, but for a regular n-gon, one has n/2 -2 special choices of l which, guarantee that every Bäcklund transform will close. This can be used to produce closed Darboux transform for isothermic surfaces that include a regular n-gon.
If the curve is parametrized by arclength (all edges have length 1), there is a S of starting points for which the new curve has constant distance from the old one.If the parameter l is real one can do this construction for curves in space too.

A regular 24-gon...	...and a closed Bäcklund transformed of it
	The straight line transforms into an elastic loop.

Author: Tim Hoffmann