Hyperbolic voronoi diagrams made easy

Proceedings - 2010 10th International Conference on Computational Science and Its Applications, ICCSA 2010

Volumn , Issue , 2010, Pages 74-80

  Nielsen, Frank ^a,b   Nock, Richard ^c  

^a ECOLE POLYTECHNIQUE   (France)

^b SONY COMPUTER SCIENCE LABORATORIES INC   (Japan)

^c UNIVERSITÉ DES ANTILLES ET DE LA GUYANE   (France)

Author keywords

Conformal non conformal geometries; Klein disk; Poincar disk; Poincar upper plane; Voronoi diagrams

Indexed keywords

CONFORMAL MAPPING; NEAREST NEIGHBOR SEARCH; USER INTERFACES;

FINITE POINT SETS; HYPERBOLIC SPACES; IMAGE CATALOGS; KLEIN DISK; MAPPING TRANSFORMATION; NEAREST NEIGHBORS; POWER DIAGRAMS; VORONOI DIAGRAMS;

COMPUTATIONAL GEOMETRY;

EID: 85006773346     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1109/ICCSA.2010.37     Document Type: Conference Paper

References (19)

1. E. Beltrami, Essay on the interpretation of the non-Euclidean geometry, 1868, translated in 1996 by J.C. Stillwell Hist Math. 10 AMS Publication.
2. J.-D. Boissonnat and M. Yvinec, Algorithmic Geometry. New York, NY, USA: Cambridge University Press, 1998.
3. K. Onishi and N. Takayama, “Construction of Voronoi diagram on the upper half-plane,” IEICE Transactions, vol. 79-A, no. 4, pp. 533–539, 1996.
4. K. Onishi and J.-I. Itoh, “Voronoi diagram in simply connected complete manifold,” IEICE Transactions, vol. E85-A, no. 5, pp. 944–948, 2002.
5. J.-D. Boissonnat, A. Cérézo, O. Devillers, and M. Teillaud, “Output-sensitive construction of the Delaunay triangulation of points lying in two planes.” Int. J. Comput. Geometry Appl., vol. 6, no. 1, pp. 1–14, 1996.
6. Z. Nilforoushan and A. Mohades, “Hyperbolic voronoi diagram.” in ICCSA (5), 2006, pp. 735–742.
7. 6: Advanced Rendering Techniques. Charles River Media, 2008.
8. S. I. R. Costa, S. A. Santos, and J. E. Strapasson, “Fisher information matrix and hyperbolic geometry,” 2005, pp. 28–30. [Online]. Available: http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1531851
9. R. E. Kass and P. W. Vos, Geometrical Foundations of Asymptotic Inference. Wiley, 1997.
10. F. Aurenhammer, “Power diagrams: Properties, algorithms and applications,” SIAM Journal of Computing, vol. 16, no. 1, pp. 78–96, 1987.
11. J.-D. Boissonnat and M. I. Karavelas, “On the combinatorial complexity of euclidean voronoi cells and convex hulls of d-dimensional spheres,” in SODA’03: Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms. Philadelphia, PA, USA: Society for Industrial and Applied Mathematics, 2003, pp. 305–312.
12. F. Nielsen, J.-D. Boissonnat, and R. Nock, “Bregman voronoi diagrams: Properties, algorithms and applications,” CoRR, vol. abs/0709.2196, 2007.
13. O. Devillers, S. Meiser, and M. Teillaud, “The space of spheres, a geometric tool to unify duality results on Voronoi diagrams,” INRIA, Research Report 1620, 1992.
14. J. A. Walter, D. Weissling, K. Essig, and H. Ritter, “Interactive hyperbolic image browsing – towards an integrated multimedia navigator,” in Proceedings of ACM KDD/MDM 2006, 2006.
15. M. W. Bern and D. Eppstein, “Optimal Möbius transformations for information visualization and meshing,” in WADS ’01: Proceedings of the 7th International Workshop on Algorithms and Data Structures. London, UK: Springer-Verlag, 2001, pp. 14–25.
16. J. Lamping, R. Rao, and P. Pirolli, “A focus+context technique based on hyperbolic geometry for visualizing large hierarchies,” in CHI’95: Proceedings of the SIGCHI conference on Human factors in computing systems. New York, NY, USA: ACM Press/Addison-Wesley Publishing Co., 1995, pp. 401–408.
17. S. Skyum, “A simple algorithm for computing the smallest enclosing circle,” Inf. Process. Lett., vol. 37, no. 3, pp. 121–125, 1991.
18. E. Welzl, “Smallest enclosing disks (balls and ellipsoids),” in New Results and New Trends in Computer Science, ser. Lecture Notes in Computer Science. Springer, 1991, vol. 555, pp. 359–370.
19. F. Nielsen and R. Nock, “On the smallest enclosing information disk,” Inf. Process. Lett., vol. 105, no. 3, pp. 93–97, 2008.