research-article

Stability of Delaunay-type structures for manifolds: [extended abstract]

Authors:

Jean-Daniel Boissonnat,

Arijit GhoshAuthors Info & Claims

SoCG '12: Proceedings of the twenty-eighth annual symposium on Computational geometry

Pages 229 - 238

https://doi.org/10.1145/2261250.2261284

Published: 17 June 2012 Publication History

Abstract

We introduce a parametrized notion of genericity for Delaunay triangulations which, in particular, implies that the Delaunay simplices of δ-generic point sets are thick. Equipped with this notion, we study the stability of Delaunay triangulations under perturbations of the metric and of the vertex positions. We then show that, for any sufficiently regular submanifold of Euclidean space, and appropriate ε and δ, any sample set which meets a localized δ-generic ε-dense sampling criteria yields a manifold intrinsic Delaunay complex which is equal to the restricted Delaunay complex.

References

[1]

P. K. Agarwal, J. Gao, L. Guibas, H. Kaplan, V. Koltun, N. Rubin, and M. Sharir. Kinetic stable Delaunay graphs. In SoCG, pages 127--136, 2010.

Digital Library

[2]

J.-D. Boissonnat, R. Dyer, and A. Ghosh. Construction of stable Delaunay structures for manifolds. Technical Report CGL-TR-??, Computational Geometric Learning, 2012. http://cgl.uni-jena.de/Publications/WebHome.

[3]

J.-D. Boissonnat, R. Dyer, and A. Ghosh. Stability of Delaunay structures for manifolds. Technical Report CGL-TR-??, Computational Geometric Learning, 2012. http://cgl.uni-jena.de/Publications/WebHome.

[4]

J.-D. Boissonnat, R. Dyer, A. Ghosh, and S. Oudot. Equating the witness and restricted Delaunay complexes. Technical Report CGL-TR-24, Computational Geometric Learning, 2011. http://cgl.uni-jena.de/Publications/WebHome.

[5]

J.-D. Boissonnat and A. Ghosh. Manifold reconstruction using tangential Delaunay complexes. Technical Report N^O 7142 v.3, INRIA, 2011.

[6]

D. Bandyopadhyay and J. Snoeyink. Almost-Delaunay simplices: nearest neighbor relations for imprecise points. In SODA, pages 410--419, 2004.

Digital Library

[7]

J.-D. Boissonnat, C. Wormser, and M. Yvinec. Anisotropic Delaunay mesh generation. Research Report RR-7712, INRIA, 2011.

[8]

S.-W. Cheng, T. K. Dey, and E. A. Ramos. Manifold reconstruction from point samples. In SODA, pages 1018--1027, 2005.

Digital Library

[9]

E. N. Dancer. Degree theory on convex sets and applications to bifurcation. In Calculus of Variations and Partial Differential Equations. pages 185--225. Springer-Verlag, 2000.

[10]

B. Delaunay. Sur la sphere video. Izv. Akad. Nauk SSSR, Otdelenie Matematicheskii i Estestvennyk Nauk, 7:793--800, 1934.

[11]

V. de Silva. A weak characterisation of the Delaunay triangulation. Geometriae Dedicata, 135:39--64, 2008.

[12]

H. Edelsbrunner and N. R. Shah. Triangulating topological spaces. Int. J. Comput. Geometry Appl., 7(4):365--378, 1997.

[13]

S. Funke, C. Klein, K. Mehlhorn, and S. Schmitt. Controlled perturbation for Delaunay triangulations. In SODA, pages 1047--1056, 2005.

Digital Library

[14]

J. Giesen and U. Wagner. Shape dimension and intrinsic metric from samples of manifolds. Discrete & Comp. Geom., 32:245--267, 2004.

Digital Library

[15]

G. Leibon and D. Letscher. Delaunay triangulations and Voronoi diagrams for Riemannian manifolds. In SoCG, pages 341--349, 2000.

Digital Library

[16]

F. Labelle and J. R. Shewchuk. Anisotropic Voronoi diagrams and guaranteed-quality anisotropic mesh generation. In SoCG, pages 191--200, 2003.

Digital Library

[17]

J. R. Munkres. Elementary differential topology. Princeton University press, second edition, 1968.

[18]

P. Niyogi, S. Smale, and S. Weinberger. Finding the homology of submanifolds with high confidence from random samples. Discrete & Comp. Geom.}, 39(1--3):419--441, 2008.

Digital Library

[19]

H. Whitney. Geometric Integration Theory. Princeton University Press, 1957.

Cited By

Devgun A(2014)A Mathematical Model to Generate 3D SurfaceProceedings of the 2014 International Conference on Computational Intelligence and Communication Networks10.1109/CICN.2014.259(1237-1242)Online publication date: 14-Nov-2014
https://dl.acm.org/doi/10.1109/CICN.2014.259
Zhong ZGuo XWang WLévy BSun FLiu YMao W(2013)Particle-based anisotropic surface meshingACM Transactions on Graphics10.1145/2461912.246194632:4(1-14)Online publication date: 21-Jul-2013
https://dl.acm.org/doi/10.1145/2461912.2461946

Stability of Delaunay-type structures for manifolds: [extended abstract]
1. Computing methodologies
  1. Computer graphics
2. Theory of computation
  1. Randomness, geometry and discrete structures

Recommendations

Manifold reconstruction using tangential Delaunay complexes
SoCG '10: Proceedings of the twenty-sixth annual symposium on Computational geometry

We give a provably correct algorithm to reconstruct a k-dimensional manifold embedded in d-dimensional Euclidean space. Input to our algorithm is a point sample coming from an unknown manifold. Our approach is based on two main ideas : the notion of ...
Delaunay/Voronoi Dual Representation of Smooth 2-Manifolds
CADGRAPHICS '11: Proceedings of the 2011 12th International Conference on Computer-Aided Design and Computer Graphics

Delaunay triangulation and Voronoi diagram are widely used in computer graphics, computational geometry and other fields. Different from the planar case, Delaunay triangulation of a point set on smooth 2-manifolds does not always exist. In this paper we ...
Delaunay Triangulation of Manifolds

We present an algorithm for producing Delaunay triangulations of manifolds. The algorithm can accommodate abstract manifolds that are not presented as submanifolds of Euclidean space. Given a set of sample points and an atlas on a compact manifold, a ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

SoCG '12: Proceedings of the twenty-eighth annual symposium on Computational geometry

June 2012

436 pages

ISBN:9781450312998

DOI:10.1145/2261250

Program Chairs:
Tamal Dey
The Ohio State University, USA
,
Sue Whitesides
University of Victoria, Canada

Copyright © 2012 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Sponsors

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 17 June 2012

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article

Conference

SoCG '12

Sponsor:

SoCG '12: Symposium on Computational Geometry 2012

June 17 - 20, 2012

North Carolina, Chapel Hill, USA

Acceptance Rates

Overall Acceptance Rate 625 of 1,685 submissions, 37%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

2
Total Citations
View Citations
132
Total Downloads

Downloads (Last 12 months)4
Downloads (Last 6 weeks)2

Reflects downloads up to 08 Mar 2025

Other Metrics

View Author Metrics

Citations

Cited By

Devgun A(2014)A Mathematical Model to Generate 3D SurfaceProceedings of the 2014 International Conference on Computational Intelligence and Communication Networks10.1109/CICN.2014.259(1237-1242)Online publication date: 14-Nov-2014
https://dl.acm.org/doi/10.1109/CICN.2014.259
Zhong ZGuo XWang WLévy BSun FLiu YMao W(2013)Particle-based anisotropic surface meshingACM Transactions on Graphics10.1145/2461912.246194632:4(1-14)Online publication date: 21-Jul-2013
https://dl.acm.org/doi/10.1145/2461912.2461946

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Figures

Tables

Media

View Table of Conten