Experimental Mathematics

Discrete periodic geodesics in a surface

Anders Linnér and Robert Renka

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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.

Article information

Experiment. Math., Volume 14, Issue 2 (2005), 145-152.

First available in Project Euclid: 30 September 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 49M25: Discrete approximations 53C22: Geodesics [See also 58E10] 58E10: Applications to the theory of geodesics (problems in one independent variable) 65K10: Optimization and variational techniques [See also 49Mxx, 93B40]

Curve-straightening elastic energy geodesic curvature periodic geodesic Sobolev gradient


Linnér, Anders; Renka, Robert. Discrete periodic geodesics in a surface. Experiment. Math. 14 (2005), no. 2, 145--152. https://projecteuclid.org/euclid.em/1128100127

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