Periodic geodesics generator. (English) Zbl 1083.53060

Linnér, Anders

Exp. Math. 13, No. 2, 199-206 (2004).

The author describes a computational algorithm for finding periodic geodesics on level surfaces. A periodic geodesic is obtained as the limit of a negative gradient trajectory of the total squared curvature functional. The algorithm exploits the inverse to the parallel transport equation that uses the geodesic curvature together with the initial point and direction to recover the curve. Several examples are given, in particular, the algorithm is illustrated in a sphere-like surface that is neither an ellipsoid, nor a surface of revolution. From computational point of view, the algorithm is adopted for Mathematica 5.0.

Reviewer: Mikhail Malakhal’tsev (Kazan)

MSC:

53C44	Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
58E10	Variational problems in applications to the theory of geodesics (problems in one independent variable)
53C22	Geodesics in global differential geometry

Keywords:

curve straightening; geodesic; gradient trajectories; steepest descent; periodic geodesic

Software:

Mathematica

References:

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