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Mathematics in Computer Science

, Volume 9, Issue 3, pp 345–353 | Cite as

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

  • M. H. M. J. WintraeckenEmail author
  • G. Vegter
Article

Abstract

Let \({\Sigma}\) be a strictly convex (hyper-)surface, S m an optimal triangulation (piecewise linear in ambient space) of \({\Sigma}\) whose m vertices lie on \({\Sigma}\) and \({\tilde{S}_m}\) an optimal triangulation of \({\Sigma}\) with m vertices. Here we use optimal in the sense of minimizing \({d_H(S_m, \Sigma)}\), where \({d_H}\) 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 \({d_H(S_m, \Sigma)}\) in m is twice that of \({d_H(\tilde{S}_m, \Sigma)}\). This statement is proven.

Keywords

Asymptotic approximations Best approximation Triangulation 

List of Symbols

Ad

Fejes Tóth’s approximation parameter

CH

Convex hull of a subset of \({\mathbb{R}^d}\)

C

Convex body

dH

Hausdorff distance

\({\kappa_d}\)

Volume of the d-dimensional ball, that is \({\pi ^{d/2} / \Gamma (1+d/2)}\).

K

Gaussian curvature

L

Line segment contained in a hypersurface

m

Number of vertices

mt

Number of triangles (only 2 dimensional case)

OLp

Hyperplane orthogonal to the line segment L going through \({p \in L}\)

Pm

Polygon/polytope with m vertices

\({P_m^{{\rm in}}}\)

Inscribed polygon/polytope with m vertices

Sm

Simplicial complex with m vertices

\({S_m^{{\rm on}}}\)

Simplicial complex whose m vertices lie on the hypersurface (often at a stage where the complex is not yet prover to be convex)

\({\Sigma}\)

Hypersurface

\({\theta_d}\)

The optimal covering density of Euclidean space by unit balls

\({U(X,\epsilon) }\)

\({\epsilon}\) neighbourhood of X

\({v_i, \tilde{v}_i}\)

Vertices

Vi

Set of vertices

Mathematics Subject Classification

Primary 41A60 41A50 32B25 Secondary 41A63 65D17 65D18 68U05 68U07 

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Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Johann Bernoulli Institute for Mathematics and Computer ScienceGroningenThe Netherlands

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