
This article is cited in 1 scientific paper (total in 1 paper)
Closed Locally Minimal Networks on the Surfaces of Convex Polyhedra
N. P. Strelkova^{} ^{} M. V. Lomonosov Moscow State University, Leninskie Gory, 1, Moscow, 119991, Russia
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 $120^{\circ}$. 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 $\frac{\pi}{3}$ have closed locally minimal networks.
Keywords:
locally minimal network, geodesic net, convex polyhedron.
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UDC:
514.113.5+514.774.8 Received: 05.08.2013
Citation:
N. P. Strelkova, “Closed Locally Minimal Networks on the Surfaces of Convex Polyhedra”, Model. Anal. Inform. Sist., 20:5 (2013), 117–147
Citation in format AMSBIB
\Bibitem{Str13}
\by N.~P.~Strelkova
\paper Closed Locally Minimal Networks on the Surfaces of Convex Polyhedra
\jour Model. Anal. Inform. Sist.
\yr 2013
\vol 20
\issue 5
\pages 117147
\mathnet{http://mi.mathnet.ru/mais336}
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http://mi.mathnet.ru/eng/mais336 http://mi.mathnet.ru/eng/mais/v20/i5/p117
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