Nice point sets can have nasty Delaunay triangulations

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Nice point sets can have nasty Delaunay triangulations

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Annual Symposium on Computational Geometry archive
Proceedings of the seventeenth annual symposium on Computational geometry table of contents

Medford, Massachusetts, United States

Pages: 96 - 105

Year of Publication: 2001

ISBN:1-58113-357-X

Author

Jeff Erickson

University of Illinois, Urbana-Champaign

Sponsors

SIGACT: ACM Special Interest Group on Algorithms and Computation Theory
SIGGRAPH: ACM Special Interest Group on Computer Graphics and Interactive Techniques

Publisher

ACM New York, NY, USA

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Downloads (6 Weeks): 3, Downloads (12 Months): 24, Citation Count: 18

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ABSTRACT

We consider the complexity of Delaunay triangulations of sets of point s in $\Real^3$ under certain practical geometric constraints. The \emph{spread} of a set of points is the ratio between the longest and shortest pairwise distances. We show that in the worst case, the Delaunay triangulation of $n$ points in~$\Real^3$ with spread $\Delta$ has complexity $\Omega(\min\set{\Delta^3, n\Delta, n^2})$ and $O(\min\set{\Delta^4, n^2})$. For the case $\Delta = \Theta(\sqrt{n})$, our lower bound construction consists of a uniform sample of a smooth convex surface with bounded curvature. We also construct a family of smooth connected surfaces such that the Delaunay triangulation of any good point sample has near-quadratic complexity.

REFERENCES

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	Sunghee Choi , Nina Amenta, Delaunay triangulation programs on surface data, Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms, p.135-136, January 06-08, 2002, San Francisco, California
	Mordecai J. Golin , Hyeon-Suk Na, On the average complexity of 3D-Voronoi diagrams of random points on convex polytopes, Computational Geometry: Theory and Applications, v.25 n.3, p.197-231, July 2003
	Leonidas Guibas , An Nguyen , Daniel Russel , Li Zhang, Collision detection for deforming necklaces, Proceedings of the eighteenth annual symposium on Computational geometry, p.33-42, June 05-07, 2002, Barcelona, Spain
	Sariel Har-Peled , Shakhar Smorodinsky, On conflict-free coloring of points and simple regions in the plane, Proceedings of the nineteenth annual symposium on Computational geometry, June 08-10, 2003, San Diego, California, USA
	Yogish Sabharwal , Nishant Sharma , Sandeep Sen, Nearest neighbors search using point location in balls with applications to approximate Voronoi decompositions, Journal of Computer and System Sciences, v.72 n.6, p.955-977, September 2006
	Sunghee Choi, The Delaunay tetrahedralization from Delaunay triangulated surfaces, Proceedings of the eighteenth annual symposium on Computational geometry, p.145-150, June 05-07, 2002, Barcelona, Spain
	Dominique Attali , Jean-Daniel Boissonnat , André Lieutier, Complexity of the delaunay triangulation of points on surfaces the smooth case, Proceedings of the nineteenth annual symposium on Computational geometry, June 08-10, 2003, San Diego, California, USA
	Pankaj Agarwal , Leonidas Guibas , An Nguyen , Daniel Russel , Li Zhang, Collision detection for deforming necklaces, Computational Geometry: Theory and Applications, v.28 n.2-3, p.137-163, June 2004
	Jeff Erickson, Local polyhedra and geometric graphs, Proceedings of the nineteenth annual symposium on Computational geometry, June 08-10, 2003, San Diego, California, USA
	Ho-Lun Cheng , Xinwei Shi, Quality mesh generation for molecular skin surfaces using restricted union of balls, Computational Geometry: Theory and Applications, v.42 n.3, p.196-206, April, 2009
	J. D. Boissonnat , S. Oudot, Provably good surface sampling and approximation, Proceedings of the 2003 Eurographics/ACM SIGGRAPH symposium on Geometry processing, June 23-25, 2003, Aachen, Germany
	Tamal K. Dey , Joachim Giesen , Samrat Goswami , Wulue Zhao, Shape dimension and approximation from samples, Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms, p.772-780, January 06-08, 2002, San Francisco, California
	Stefan Funke , Edgar A. Ramos, Smooth-surface reconstruction in near-linear time, Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms, p.781-790, January 06-08, 2002, San Francisco, California
	Siu-Wing Cheng , Tamal K. Dey , Edgar A. Ramos , Tathagata Ray, Sampling and meshing a surface with guaranteed topology and geometry, Proceedings of the twentieth annual symposium on Computational geometry, June 08-11, 2004, Brooklyn, New York, USA
	Dominique Attali , Jean-Daniel Boissonnat, A linear bound on the complexity of the delaunay triangulation of points on polyhedral surfaces, Proceedings of the seventh ACM symposium on Solid modeling and applications, June 17-21, 2002, Saarbrücken, Germany
	Raphaëlle Chaine, A geometric convection approach of 3-D reconstruction, Proceedings of the 2003 Eurographics/ACM SIGGRAPH symposium on Geometry processing, June 23-25, 2003, Aachen, Germany
	Jeff Erickson, Dense point sets have sparse Delaunay triangulations: or "…but not too nasty", Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms, p.125-134, January 06-08, 2002, San Francisco, California