Closed Locally Minimal Networks on the Surfaces of Convex Polyhedra
https://doi.org/10.18255/1818-1015-2013-5-117-147
Abstract
Closed locally minimal networks can be viewed as “branching” closed geodesics. We study such networks on the surfaces of convex polyhedra and discuss the problem of describing the set of all convex polyhedra that have such networks. A closed locally minimal network on a convex polyhedron is an embedding of a graph provided that all edges are geodesic arcs and at each vertex exactly three adges meet at angles of 120∘ . In this paper, we do not deal with closed (periodic) geodesics. Among other results, we prove that the natural condition on the curvatures of a polyhedron that is necessary for the polyhedron to have a closed locally minimal network on its surface is not sufficient. We also prove a new stronger necessary condition. We describe all possible combinatorial structures and edge lengths of closed locally minimal networks on convex polyhedra. We prove that almost all convex polyhedra with vertex curvatures divisible by π/3 have closed locally minimal networks.
About the Author
N. P. Strelkova
M.V. Lomonosov Moscow State University;
ЯрГУ им. П.Г. Демидова
Russian Federation
Russian Federation
аспирант, Leninskie Gory, 1, Moscow, 119991, Russia;
лаборант Международной лаборатории “Дискретная и вычислительная геометрия” им. Б.Н. Делоне
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Review
For citations:
Strelkova N.P. Closed Locally Minimal Networks on the Surfaces of Convex Polyhedra. Modeling and Analysis of Information Systems. 2013;20(5):117-147. (In Russ.) https://doi.org/10.18255/1818-1015-2013-5-117-147
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