Article

An empirical comparison of techniques for updating Delaunay triangulations

Authors:

Leonidas Guibas,

Daniel RusselAuthors Info & Claims

SCG '04: Proceedings of the twentieth annual symposium on Computational geometry

Pages 170 - 179

https://doi.org/10.1145/997817.997846

Published: 08 June 2004 Publication History

Get Access

Abstract

The computation of Delaunay triangulations from static point sets has been extensively studied in computational geometry. When the points move with known trajectories, kinetic data structures can be used to maintain the triangulation. However, there has been little work so far on how to maintain the triangulation when the points move without explicit motion plans, as in the case of a physical simulation. In this paper we examine how to update Delaunay triangulations after small displacements of the defining points, as might be provided by a physics-based integrator. We have implemented a variety of update algorithms, many new, toward this purpose. We ran these algorithms on a corpus of data sets to provide running time comparisons and determined that updating Delaunay can be significantly faster than recomputing.

References

[1]

M. Abellanas, F. Hurtado, and P. A. Ramos. Structural tolerance and Delaunay triangulation. Information Processing Let-ters, 71(5-6):221--227, 1999.

Digital Library

Google Scholar

[2]

N. Amenta and M. W. Bern. Surface reconstruction by voronoi filtering. In Symp. on Comp. Geom., pages 39--48, 1998.

Digital Library

Google Scholar

[3]

N. Amenta, S. Choi, and G. Rote. Incremental constructions con BRIO. In Proc. of the 19th conference on Comp. geometry, pages 211--219, 2003.

Digital Library

Google Scholar

[4]

D. Attali, J.-D. Boissonnat, and A. Lieutier. Complexity of the delaunay triangulation of points on surfaces the smooth case. In Proc. of the 19th conference on Comp. geometry, pages 201--210. ACM Press, 2003.

Digital Library

Google Scholar

[5]

F. Aurenhammer. Power diagrams: properties, algorithms, and applications. SIAM J. on Computing, 16:78--96, 1987.

Digital Library

Google Scholar

[6]

D. Bandyopadhyay and J. Snoeyink. Almost-Delaunay simplices: Nearest neighbor relations for imprecise points. In ACM-SIAM Symp. on Discrete Algorithms, pages 403--412, 2004.

Digital Library

Google Scholar

[7]

J. Basch, L. Guibas, and J. Hershberger. Data structures for mobile data. In ACM-SIAM Symp. on Discrete Algorithms, pages 747--756, 1997.

Digital Library

Google Scholar

[8]

J. Basch, L. J. Guibas, and J. Hershberger. Data structures for mobile data. In ACM-SIAM symp. on discrete algorithms, pages 747--756. Society for Industrial and Applied Mathematics, 1997.

Digital Library

Google Scholar

[9]

J. Basch, L. J. Guibas, C. D. Silverstein, and L. Zhang. A practical evaluation of kinetic data structures. In Proc. of the 13th annual symp. on Comp. geometry, pages 388--390, 1997.

Digital Library

Google Scholar

[10]

P. Cignoni, C. Montani, and R. Scopigno. A fast divide and conquer Delaunay triangulation algorithm in Ed. Computer Aided Design, 30:333--341, 1998.

Crossref

Google Scholar

[11]

O. Devillers. Improved incremental randomized Delaunay triangulation. In Proc. of the 14th annual symp. on comp. geometry, 1998.

Digital Library

Google Scholar

[12]

O. Devillers and S. Pion. Efficient exact geometric predicates for delaunay triangulations. In Proc. 5th Workshop Algorithm Engineering Experiments, pages 37--44, 2003.

Google Scholar

[13]

O. Devillers and M. Teillaud. Perturbations and vertex removal in a 3d Delaunay triangulation. In ACM-SIAM symp. on Discrete algorithms, pages 313--319. Society for Industrial and Applied Mathematics, 2003.

Digital Library

Google Scholar

[14]

H. Edelsbrunner. The union of balls and its dual shape. Discrete Comp. Geom., 13:415--440, 1995.

Digital Library

Google Scholar

[15]

H. Edelsbrunner and P. Koehl. The weighted volume derivative of a space-filling diagram. In Proc. of the National Academy of Science, volume 100, pages 2203--2208, 2003.

Crossref

Google Scholar

[16]

H. Edelsbrunner and R. Seidel. Voronoi diagrams and arrangements. In Proc. of the 1st annual symp. on comp. geometry, pages 251--262. ACM Press, 1985.

Digital Library

Google Scholar

[17]

H. Edelsbrunner and N. R. Shah. Incremental topological flipping works for regular triangulations. In Proc. of the 8th annual symp. on comp. geometry, pages 43--52. ACM Press, 1992.

Digital Library

Google Scholar

[18]

J. Erickson. Local polyhedra and geometric graphs. In Proc. of the 19th annual conference on Comp. geometry, pages 171--180. ACM Press, 2003.

Digital Library

Google Scholar

[19]

J. Erikson. Dense point sets have sparse Delaunay triangulations. In ACM-SIAM Symp. on Discrete Algorithms, pages 125--134, 2002.

Digital Library

Google Scholar

[20]

G. D. Fabritiis and P. V. Coveney. Dynamical geometry for multiscale dissipative particle dynamics. arXiv:cond-mat: 0301378.

Google Scholar

[21]

J. Gao, L. Guibas, and A. Nguyen. Deformable spanners and their applications. 2004.

Google Scholar

[22]

M. Gavrilova and J. Rokne. Collision detection optimization in a multi-particle system. International Workshop on Comp. Geom. and Applications, 2331:105--114, 2002.

Digital Library

Google Scholar

[23]

GNU scientific library. http://www.gnu.org/software/gsl/.

Google Scholar

[24]

E. Guendelman, R. Bridson, and R. Fedkiw. Nonconvex rigid bodies with stacking. ACM Transactions on Graphics, 22(3):871--878, 2003.

Digital Library

Google Scholar

[25]

L. Guibas. Kinetic data structures: A state of the art report. In Proc. 3rd Workshop on Algorithmic Foundations of Robotics, 1998.

Digital Library

Google Scholar

[26]

L. Guibas and M. Karavelas. Interval methods for kinetic simulations. In Proc. of the 15th annual symp. on comp. geometry, pages 255--264, 1999.

Digital Library

Google Scholar

[27]

L. Guibas, M. Karaveles, and D. Russel. A comp. framework for handling motion. In ALENEX 2004, Lecture notes in Computer science, 2004. to appear.

Google Scholar

[28]

L. Guibas, D. Knuth, and M. Sharir. Randomized incremental construction of Delaunay and Voronoi diagrams. Algorithmica, 7:381--413, 1992.

Digital Library

Google Scholar

[29]

L. Guibas, A. Nguyen, D. Russel, and L. Zhang. Collision detection for deforming necklaces. In Proc. of the 18th annual symp. on comp. geom, pages 33--42. ACM Press, 2002.

Digital Library

Google Scholar

[30]

L. J. Guibas and L. Zhang. Euclidean proximity and power diagrams. In Canadian Conference on Comp. Geom., pages 90--91, 1998.

Google Scholar

[31]

B. Joe. Three-dimensional triangulations from local transformations. SIAM J. on Scientific and Statistical Computing, 10:718--741, 1989.

Digital Library

Google Scholar

[32]

J. R. Shewchuk. A condition guaranteeing the existence of higher-dimensional constrained Delaunay triangulations. In Proc. of the 14th annual symp. on Comp. geometry, pages 76--85, 1998.

Digital Library

Google Scholar

[33]

P. Su and R. L. D. III. A comparison of sequential Delaunay triangulation algorithms. In Proc. of the 11th annual symp. on Comp. Geom., pages 61--70, 1995.

Digital Library

Google Scholar

[34]

Tinker molecular simulation package. http://dasher.wustl.edu/tinker/.

Google Scholar

[35]

R. C. Whaley, A. Petitet, and J. Dongarra. Automated empirical optimizations of software and the ATLAS project. Parallel Computing, 27(1-2):3--35, 2001.

Digital Library

Google Scholar

Cited By

View all

Kolingerová IVomáčka TMaňák MFerko A(2018)Neighbourhood Graphs and Locally Minimal TriangulationsTransactions on Computational Science XXXIII10.1007/978-3-662-58039-4_7(115-127)Online publication date: 16-Sep-2018
https://doi.org/10.1007/978-3-662-58039-4_7
Sun WTsiotras P(2017)Sequential pursuit of multiple targets under external disturbances via Zermelo–Voronoi diagramsAutomatica10.1016/j.automatica.2017.03.01581(253-260)Online publication date: Jul-2017
https://doi.org/10.1016/j.automatica.2017.03.015
Kolingerová IFerko AVomáčka TMaňák M(2017)Nearest Neighbour Graph and Locally Minimal TriangulationComputational Science and Its Applications – ICCSA 201710.1007/978-3-319-62395-5_31(455-464)Online publication date: 7-Jul-2017
https://doi.org/10.1007/978-3-319-62395-5_31
Show More Cited By

Index Terms

An empirical comparison of techniques for updating Delaunay triangulations
1. Computing methodologies
  1. Computer graphics
    1. Shape modeling
2. Theory of computation
  1. Design and analysis of algorithms
  2. Randomness, geometry and discrete structures
    1. Computational geometry

Recommendations

Conforming weighted delaunay triangulations

Given a set of points together with a set of simplices we show how to compute weights associated with the points such that the weighted Delaunay triangulation of the point set contains the simplices, if possible. For a given triangulated surface, this ...
Flips in Higher Order Delaunay Triangulations
LATIN 2020: Theoretical Informatics
Abstract
We study the flip graph of higher order Delaunay triangulations. A triangulation of a set S of n points in the plane is order-k Delaunay if for every triangle its circumcircle encloses at most k points of S. The flip graph of S has one vertex for ...
Incrementally constructing and updating constrained Delaunay tetrahedralizations with finite-precision coordinates

Constrained Delaunay tetrahedralizations (CDTs) are valuable for discretizing three-dimensional domains with constraints such as edges and polygons. But they are difficult to generate and maintain robustly when finite-precision coordinates yield ...

Reviews

Reviewer: Patrick J. Ryan

Efficient techniques for computing the Voronoi diagram and Delaunay triangulation for a finite set of points in Euclidean space E d are well known. This paper is concerned with the problem of computing the Delaunay triangulation D(P') of a perturbed point set P' when the original point set P and its Delaunay triangulation D(P) are given. Of course, an underlying assumption is that the amount by which P' differs from P is small, so that D(P') can be expected to share much of its combinatorial and topological structure with D(P) . As the title indicates, this paper is a report on empirical research comparing various algorithms for solving this problem in E 3. Two basic approaches are compared: kinetic and static. In the kinetic approach, a specific interpolation method is chosen, and the data structure is updated at events, occurring when some pair of adjacent cells no longer satisfies the necessary local criterion for being Delaunay (the "InCircle" condition, though actually "InSphere" would be more appropriate terminology in three dimensions). In the static approach, only the initial and final configurations are considered, and the algorithms use some combination of point removal and flipping on pairs of adjacent cells that do not satisfy InCircle. For each approach, various ways of setting up the algorithms are discussed (for example, how to interpolate, and when to use point removal or flipping). These are rather technical, and are not fully specified in the paper, but relevant references to other sources are given. Readers who are familiar with such techniques will find the results of the authors' extensive tests interesting. The authors discuss which parts of the computations are most costly, and where improvements might be possible. Their results show that, generally speaking, the kinetic approaches are not currently competitive with the static approaches. Online Computing Reviews Service

Access critical reviews of Computing literature here

Become a reviewer for Computing Reviews.

Comments

Information & Contributors

Information

Published In

SCG '04: Proceedings of the twentieth annual symposium on Computational geometry

June 2004

468 pages

ISBN:1581138857

DOI:10.1145/997817

Conference Chairs:
Jack Snoeyink
UNC Chapel Hill
,
Jean-Daniel Boissonnat
INRIA Sophia Antipolis

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]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 08 June 2004

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tag

Delaunay triangulation update motion

Qualifiers

Article

Conference

SoCG04

Sponsor:

SoCG04: Annual ACM Symposium on Computational Geometry 2004

June 8 - 11, 2004

New York, Brooklyn, USA

Acceptance Rates

SCG '04 Paper Acceptance Rate 49 of 147 submissions, 33%;

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

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

29
Total Citations
View Citations
906
Total Downloads

Downloads (Last 12 months)6
Downloads (Last 6 weeks)1

Reflects downloads up to 06 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

View all

Kolingerová IVomáčka TMaňák MFerko A(2018)Neighbourhood Graphs and Locally Minimal TriangulationsTransactions on Computational Science XXXIII10.1007/978-3-662-58039-4_7(115-127)Online publication date: 16-Sep-2018
https://doi.org/10.1007/978-3-662-58039-4_7
Sun WTsiotras P(2017)Sequential pursuit of multiple targets under external disturbances via Zermelo–Voronoi diagramsAutomatica10.1016/j.automatica.2017.03.01581(253-260)Online publication date: Jul-2017
https://doi.org/10.1016/j.automatica.2017.03.015
Kolingerová IFerko AVomáčka TMaňák M(2017)Nearest Neighbour Graph and Locally Minimal TriangulationComputational Science and Its Applications – ICCSA 201710.1007/978-3-319-62395-5_31(455-464)Online publication date: 7-Jul-2017
https://doi.org/10.1007/978-3-319-62395-5_31
Smolik MSkala V(2015)Highly Parallel Algorithm for Large Data InCore and OutCore Triangulation in E2 and E3Procedia Computer Science10.1016/j.procs.2015.05.36951:C(2613-2622)Online publication date: 1-Sep-2015
https://dl.acm.org/doi/10.1016/j.procs.2015.05.369
Cao TNanjappa AGao MTan TKeyser JSander P(2014)A GPU accelerated algorithm for 3D Delaunay triangulationProceedings of the 18th meeting of the ACM SIGGRAPH Symposium on Interactive 3D Graphics and Games10.1145/2556700.2556710(47-54)Online publication date: 14-Mar-2014
https://dl.acm.org/doi/10.1145/2556700.2556710
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
Krijgsman DOgarko VLuding S(2014)Optimal parameters for a hierarchical grid data structure for contact detection in arbitrarily polydisperse particle systemsComputational Particle Mechanics10.1007/s40571-014-0020-91:3(357-372)Online publication date: 22-May-2014
https://doi.org/10.1007/s40571-014-0020-9
Smolik MSkala V(2014)Fast Parallel Triangulation Algorithm of Large Data Sets in E2 and E3 for In-Core and Out-Core Memory ProcessingComputational Science and Its Applications – ICCSA 201410.1007/978-3-319-09129-7_23(301-314)Online publication date: 2014
https://doi.org/10.1007/978-3-319-09129-7_23
Wei Sun Tsiotras P(2013)A sequential pursuer-target assignment problem under external disturbances52nd IEEE Conference on Decision and Control10.1109/CDC.2013.6760500(3994-3999)Online publication date: Dec-2013
https://doi.org/10.1109/CDC.2013.6760500
Bouza-Rodríguez BComesaña-Campos AMenéndez-Díaz AGarcia-Cortes S(2013)A novel geometric approach for 3-D geological modellingUne nouvelle approche géométrique pour la modélisation géologique 3-DBulletin of Engineering Geology and the Environment10.1007/s10064-013-0545-973:2(551-567)Online publication date: 16-Nov-2013
https://doi.org/10.1007/s10064-013-0545-9
Show More Cited By

View Options

Login options

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

Cited By

Index Terms

Recommendations

Conforming weighted delaunay triangulations

Flips in Higher Order Delaunay Triangulations

Incrementally constructing and updating constrained Delaunay tetrahedralizations with finite-precision coordinates

Reviews

Access critical reviews of Computing literature here

Comments

Published In

Sponsors

Publisher

Publication History

Permissions

Check for updates

Author Tag

Qualifiers

Conference

Acceptance Rates

Other Metrics

Article Metrics

Other Metrics

Cited By

Login options

Full Access

PDF

eReader

Abstract

References

Cited By

Index Terms

Recommendations

Conforming weighted delaunay triangulations

Flips in Higher Order Delaunay Triangulations

Incrementally constructing and updating constrained Delaunay tetrahedralizations with finite-precision coordinates

Reviews

Access critical reviews of Computing literature here

Comments

Information

Published In

Sponsors

Publisher

Publication History

Permissions

Check for updates

Author Tag

Qualifiers

Conference

Acceptance Rates

Contributors

Other Metrics

Bibliometrics

Article Metrics

Other Metrics

Citations

Cited By

Login options

Full Access

View options

PDF

eReader

Figures

Other

Share

Share this Publication link

Share on social media

Affiliations