Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

On the Optimal Triangulation of Convex Hypersurfaces, Whose Vertices Lie in Ambient Space

On the Optimal Triangulation of Convex Hypersurfaces, Whose Vertices Lie in Ambient Space Let $${\Sigma}$$ Σ be a strictly convex (hyper-)surface, S m an optimal triangulation (piecewise linear in ambient space) of $${\Sigma}$$ Σ whose m vertices lie on $${\Sigma}$$ Σ and $${\tilde{S}_m}$$ S ~ m an optimal triangulation of $${\Sigma}$$ Σ with m vertices. Here we use optimal in the sense of minimizing $${d_H(S_m, \Sigma)}$$ d H ( S m , Σ ) , where $${d_H}$$ d H denotes the Hausdorff distance. In ‘Lagerungen in der Ebene, auf der Kugel und im Raum’ Fejes Tóth conjectured that the leading term in the asymptotic development of $${d_H(S_m, \Sigma)}$$ d H ( S m , Σ ) in m is twice that of $${d_H(\tilde{S}_m, \Sigma)}$$ d H ( S ~ m , Σ ) . This statement is proven. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematics in Computer Science Springer Journals

On the Optimal Triangulation of Convex Hypersurfaces, Whose Vertices Lie in Ambient Space

Download PDF
9 pages

Loading next page...
 
/lp/springer-journals/on-the-optimal-triangulation-of-convex-hypersurfaces-whose-vertices-0uYgo1RGMZ

References (7)

Publisher
Springer Journals
Copyright
Copyright © 2014 by Springer Basel
Subject
Mathematics; Mathematics, general; Computer Science, general
ISSN
1661-8270
eISSN
1661-8289
DOI
10.1007/s11786-014-0216-7
Publisher site
See Article on Publisher Site

Abstract

Let $${\Sigma}$$ Σ be a strictly convex (hyper-)surface, S m an optimal triangulation (piecewise linear in ambient space) of $${\Sigma}$$ Σ whose m vertices lie on $${\Sigma}$$ Σ and $${\tilde{S}_m}$$ S ~ m an optimal triangulation of $${\Sigma}$$ Σ with m vertices. Here we use optimal in the sense of minimizing $${d_H(S_m, \Sigma)}$$ d H ( S m , Σ ) , where $${d_H}$$ d H denotes the Hausdorff distance. In ‘Lagerungen in der Ebene, auf der Kugel und im Raum’ Fejes Tóth conjectured that the leading term in the asymptotic development of $${d_H(S_m, \Sigma)}$$ d H ( S m , Σ ) in m is twice that of $${d_H(\tilde{S}_m, \Sigma)}$$ d H ( S ~ m , Σ ) . This statement is proven.

Journal

Mathematics in Computer Science – Springer Journals

Published: Nov 14, 2014

There are no references for this article.