Closed Locally Minimal Networks on the Surfaces of Convex Polyhedra
<p>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...
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| Format: | Article |
| Language: | English |
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Yaroslavl State University
2013-01-01
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| Series: | Моделирование и анализ информационных систем |
| Subjects: | |
| Online Access: | http://mais-journal.ru/jour/article/view/178 |
| _version_ | 1828076954593525760 |
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| author | N. P. Strelkova |
| author_facet | N. P. Strelkova |
| author_sort | N. P. Strelkova |
| collection | DOAJ |
| description | <p>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.</p> |
| first_indexed | 2024-04-11T02:22:27Z |
| format | Article |
| id | doaj.art-3744b80b09a84770a8dbfb46127c7e97 |
| institution | Directory Open Access Journal |
| issn | 1818-1015 2313-5417 |
| language | English |
| last_indexed | 2024-04-11T02:22:27Z |
| publishDate | 2013-01-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj.art-3744b80b09a84770a8dbfb46127c7e972023-01-02T23:20:38ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172013-01-01205117147172Closed Locally Minimal Networks on the Surfaces of Convex PolyhedraN. P. Strelkova0МГУ им. М. В. Ломоносова; ЯрГУ им. П.Г. Демидова<p>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.</p>http://mais-journal.ru/jour/article/view/178локально минимальная сетьгеодезическая сетьвыпуклый многогранник |
| spellingShingle | N. P. Strelkova Closed Locally Minimal Networks on the Surfaces of Convex Polyhedra Моделирование и анализ информационных систем локально минимальная сеть геодезическая сеть выпуклый многогранник |
| title | Closed Locally Minimal Networks on the Surfaces of Convex Polyhedra |
| title_full | Closed Locally Minimal Networks on the Surfaces of Convex Polyhedra |
| title_fullStr | Closed Locally Minimal Networks on the Surfaces of Convex Polyhedra |
| title_full_unstemmed | Closed Locally Minimal Networks on the Surfaces of Convex Polyhedra |
| title_short | Closed Locally Minimal Networks on the Surfaces of Convex Polyhedra |
| title_sort | closed locally minimal networks on the surfaces of convex polyhedra |
| topic | локально минимальная сеть геодезическая сеть выпуклый многогранник |
| url | http://mais-journal.ru/jour/article/view/178 |
| work_keys_str_mv | AT npstrelkova closedlocallyminimalnetworksonthesurfacesofconvexpolyhedra |