Discrete periodic geodesics in a surface

Abstract

An alternative to the traditional curve-straightening flow on periodic curves in surfaces is introduced. The implementation of this flow produces periodic geodesics in minutes rather than hours. The flow is also simpler to initiate since its use of a penalty method permits initial curves that are not necessarily in the surface. Compact and noncompact examples are provided as well as examples with trivial and nontrivial free homotopy classes. The explicit curve-straightening flow on circles in Euclidian space is derived to help check the consistency of the implementations.