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On the exact maximum complexity of Minkowski sums of convex polyhedra

Authors:

Efi Fogel,

Dan Halperin,

Christophe WeibelAuthors Info & Claims

SCG '07: Proceedings of the twenty-third annual symposium on Computational geometry

Pages 319 - 326

https://doi.org/10.1145/1247069.1247126

Published: 06 June 2007 Publication History

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References

[1]

{1} P. K. Agarwal, E. Flato, and D. Halperin. Polygon decomposition for efficient construction of Minkowski sums. Comput. Geom. Theory Appl., 21:39-61, 2002.

Digital Library

Google Scholar

[2]

{2} J. Basch, L. J. Guibas, G. D. Ramkumar, and L. Ramshaw. Polyhedral tracings and their convolution. In Proc. 2nd Workshop Alg. Found. Robot., 1996.

Google Scholar

[3]

{3} H. Bekker and J. B. T. M. Roerdink. An efficient algorithm to calculate the Minkowski sum of convex 3D polyhedra. In Proc. of the Int. Conf. on Comput. Sci.-Part I, pages 619-628. Springer-Verlag, 2001.

Digital Library

Google Scholar

[4]

{4} J.-D. Boissonnat, E. de Lange, and M. Teillaud. Minkowski operations for satellite antenna layout. In Proc. 13th Annu. ACM Sympos. on Comput. Geom., pages 67-76, 1997.

Digital Library

Google Scholar

[5]

{5} S. A. Cameron and R. K. Culley. Determining the minimum translational distance between two convex polyhedra. In Proc. of IEEE Int. Conf. Robot. Auto., pages 591-596, 1986.

Crossref

Google Scholar

[6]

{6} E. Fogel and D. Halperin. Exact and efficient construction of Minkowski sums of convex polyhedra with applications. In Proc. 8th Workshop Alg. Eng. Exper. (Alenex'06), 2006.

Crossref

Google Scholar

[7]

{7} K. Fukuda. From the zonotope construction to the Minkowski addition of convex polytopes. Journal of Symbolic Computation, 38(4):1261-1272, 2004.

Digital Library

Google Scholar

[8]

{8} K. Fukuda and C. Weibel. On f-vectors of Minkowski additions of convex polytopes. Discrete and Computational Geometry. To appear.

Digital Library

Google Scholar

[9]

{9} P. K. Ghosh. A unified computational framework for Minkowski operations. Comp. Graph., 17(4):357-378, 1993.

Crossref

Google Scholar

[10]

{10} M. Granados, P. Hachenberger, S. Hert, L. Kettner, K. Mehlhorn, and M. Seel. Boolean operations on 3D selective Nef complexes: Data structure, algorithms, and implementation. In Proc. 11th Annu. Euro. Sympos. Alg., volume 2832 of LNCS, pages 174-186. Springer-Verlag, 2003.

Google Scholar

[11]

{11} P. Gritzmann and B. Sturmfels. Minkowski addition of polytopes: Computational complexity and applications to Gröbner bases. SIAM J. Disc. Math, 6(2):246-269, 1993.

Digital Library

Google Scholar

[12]

{12} L. J. Guibas, L. Ramshaw, and J. Stolfi. A kinetic framework for computational geometry. In Proc. 24th Annu. IEEE Sympos. Found. Comput. Sci., pages 100-111, 1983.

Digital Library

Google Scholar

[13]

{13} D. Halperin, L. Kavraki, and J.-C. Latombe. Robotics. In J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, 2nd Edition, chapter 48, pages 1065-1093. CRC, 2004.

Digital Library

Google Scholar

[14]

{14} C. D. Hodgson, I. Rivin, and W. D. Smith. A characterization of convex hyperbolic polyhedra and of convex polyhedra inscribed in the sphere. Bull. (New Series) of the AMS, 27:246-251, 1992.

Google Scholar

[15]

{15} A. Kaul and J. Rossignac. Solid-interpolation deformations: Construction and animation of PIPs. In Eurographics'91, pages 493-505, 1991.

Google Scholar

[16]

{16} M. C. Lin and D. Manocha. Collision and proximity queries. In J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, 2nd Edition, chapter 35, pages 787-807. CRC, 2004.

Google Scholar

[17]

{17} M. Sharir. Algorithmic motion planning. In J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, 2nd Edition, chapter 47, pages 1037-1064. CRC, 2004.

Digital Library

Google Scholar

[18]

{18} C. Weibel. Minkowski sums. http://roso.epfl.ch/cw/poly/public.php.

Google Scholar

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Papadimitriou DSojoudi S(2022)Control and Uncertainty Propagation in the Presence of Outliers by Utilizing Student-t Process Regression2022 American Control Conference (ACC)10.23919/ACC53348.2022.9867326(2748-2754)Online publication date: 8-Jun-2022
https://doi.org/10.23919/ACC53348.2022.9867326
Fogel EHalperin D(2018)Exact and efficient construction of Minkowski sums of convex polyhedra with applicationsComputer-Aided Design10.1016/j.cad.2007.05.01739:11(929-940)Online publication date: 30-Dec-2018
https://dl.acm.org/doi/10.1016/j.cad.2007.05.017
Fogel EHalperin D(2009)Polyhedral Assembly Partitioning with Infinite Translations or The Importance of Being ExactAlgorithmic Foundation of Robotics VIII10.1007/978-3-642-00312-7_26(417-432)Online publication date: 2009
https://doi.org/10.1007/978-3-642-00312-7_26
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Index Terms

On the exact maximum complexity of Minkowski sums of convex polyhedra
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
      1. Enumeration

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Published In

SCG '07: Proceedings of the twenty-third annual symposium on Computational geometry

June 2007

404 pages

ISBN:9781595937056

DOI:10.1145/1247069

Program Chair:
Jeff Erickson
University of Illinois, Urbana-Champaign, USA

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]

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Published: 06 June 2007

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Cited By

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Papadimitriou DSojoudi S(2022)Control and Uncertainty Propagation in the Presence of Outliers by Utilizing Student-t Process Regression2022 American Control Conference (ACC)10.23919/ACC53348.2022.9867326(2748-2754)Online publication date: 8-Jun-2022
https://doi.org/10.23919/ACC53348.2022.9867326
Fogel EHalperin D(2018)Exact and efficient construction of Minkowski sums of convex polyhedra with applicationsComputer-Aided Design10.1016/j.cad.2007.05.01739:11(929-940)Online publication date: 30-Dec-2018
https://dl.acm.org/doi/10.1016/j.cad.2007.05.017
Fogel EHalperin D(2009)Polyhedral Assembly Partitioning with Infinite Translations or The Importance of Being ExactAlgorithmic Foundation of Robotics VIII10.1007/978-3-642-00312-7_26(417-432)Online publication date: 2009
https://doi.org/10.1007/978-3-642-00312-7_26
Guo XXie LGao Y(2008)Optimal Accurate Minkowski Sum Approximation of Polyhedral ModelsProceedings of the 4th international conference on Intelligent Computing: Advanced Intelligent Computing Theories and Applications - with Aspects of Theoretical and Methodological Issues10.1007/978-3-540-87442-3_23(179-188)Online publication date: 15-Sep-2008
https://dl.acm.org/doi/10.1007/978-3-540-87442-3_23

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Exact minkowski sums of convex polyhedra

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