Computing the Voronoi cells of planes, spheres and cylinders in R3

Computer Aided Geometric Design

Volumn 26, Issue 6, 2009, Pages 695-710

  Hanniel, Iddo ^a   Elber, Gershon ^a  

^a TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

Author keywords

Constructive solid geometry; Medial axis; Skeleton

Indexed keywords

CONIC SECTIONS; CONSTRUCTIVE SOLID GEOMETRY; LOWER ENVELOPES; MEDIAL AXIS; QUADRIC SURFACES; SKELETON; VORONOI CELLS;

CYLINDERS (SHAPES);

EID: 67349222521     PISSN: 01678396     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cagd.2008.09.010     Document Type: Article

References (41)

1. Amenta, N., Choi, S., Kolluri, R., 2001. The power crust. In: Proceedings of the ACM Symposium on Solid Modeling and Applications, pp. 139-146
2. Armstrong C.G. Modeling requirements for finite element analysis. Computer Aided Design 26 7 (1994) 573-578
3. Aurenhammer F., and Imai H. Geometric relations among Voronoi diagrams. STACS 87, 4th Annual Symposium on Theoretical Aspects of Computer Science. LNCS vol. 247 (1987), Springer-Verlag 53-65
4. Austern M.H. Generic Programming and the Stl: Using and Extending the C++ Standard Template Library (1998), Addison-Wesley
5. Bajaj G.S., and Thiel E. (3-4)-weighted skeleton decomposition for pattern representation and description. Pattern Recognition 27 8 (1994) 1039-1049
6. Berberich, E., Hemmer, M., Kettner, L., Schömer, E., Wolpert, N., 2005. An exact, complete and efficient implementation for computing planar maps of quadric intersection curves. In: Proceedings of the 21st ACM Symposium on Computational Geometry, pp. 99-106
7. Blum H. Biological shape and visual science (Part I). Journal of Theoretical Biology 38 (1973) 205-287
8. Boissonnat J.-D., and Delage C. Convex hull and Voronoi diagram of additively weighted points. Proceedings of the 13th Annual European Symposium on Algorithms (ESA). LNCS vol. 3669 (2005), Springer-Verlag 367-378
9. Boissonnat J.-D., and Yvinec M. Algorithmic Geometry (1998), Cambridge University Press, UK. http://cgal.inria.fr/Publications/1998/BY98 Translated from French by Hervé Brönnimann
10. Cohen E., Riesenfeld R.F., and Elber G. Geometric Modeling with Splines: An Introduction (2001), A. K. Peters
11. de Berg M., van Kreveld M., Overmars M., and Schwarzkopf O. Computational Geometry, Algorithms and Applications. 2nd ed (1998), Springer, New York
12. Dutta D., and Hoffmann C.M. On the skeleton of simple CSG objects. ASME Transactions: Journal of Mechanical Design 115 (1993) 87-94
13. Dutta D., Martin R., and Pratt M. Cyclides in surface and solid modeling. IEEE Computer Graphics and Applications 13 1 (1993) 53-59
14. Elber G. The IRIT 9.5 User Manual. http://www.cs.technion.ac.il/~irit
15. Elber G., and Kim M.-S. Computing rational bisectors. IEEE Computer Graphics and Applications 19 6 (1999)
16. Elber, G., Kim, M.S., 1999b. Rational bisectors of CSG primitives. In: Proceedings of the ACM Symposium on Solid Modeling and Applications, pp. 159-166
17. Elber, G., Kim, M.-S., 2000. A computational model for nonrational bisector surfaces: Curve-surface and surface-surface bisectors. In: Proceedings of Geometric Modeling and Processing 2000 (GMP2000), pp. 364-372
18. Elber, G., Kim, M.-S., 2001. Geometric constraint solver using multivariate rational spline functions. In: Proceedings of the ACM Symposium on Solid Modeling and Applications, pp. 1-10
19. Emiris, I.Z., Tsigaridas, E.P., Tzoumas, G.M., 2006. The predicates for the Voronoi diagram of ellipses. In: Proceedings of the 22nd ACM Symposium on Computational Geometry, pp. 227-236
20. Etzion, M., Rappoport, S.A., 1999. Computing the Voronoi diagram of a 3-d polyhedron by separate computation of its symbolic and geometric parts. In: Proceedings of the ACM Symposium on Solid Modeling and Applications, pp. 167-178
21. Fabri A., Giezeman G.-J., Kettner L., Schirra S., and Schönherr S. On the design of cgal a computational geometry algorithms library. Software - Practice and Experience 30 11 (2000) 1167-1202. citeseer.ist.psu.edu/fabri99design.html
22. Foskey, M., Lin, M., Manocha, D., 2003. Efficient computation of a simplified medial axis. In: Proceedings of the ACM Symposium on Solid Modeling and Applications, pp. 96-107
23. Hanniel, I., Ramanathan, M., Elber, G., Kim, M.-S., 2005. Precise Voronoi cell extraction of freeform rational planar closed curves. In: Shapiro, V., Kobbelt, L. (Eds.), Proceedings of the ACM Symposium on Solid and Physical Modeling, pp. 51-59
24. Held M. Voronoi diagram and offset curves of curvilinear polygons. Computer-Aided Design 30 4 (1998) 287-300
25. Hirst A. Blending cones and planes by using cyclides of Dupin. Bulletin of the Institute of Mathematics and its Applications 26 3 (1990) 41-46
26. Hoff, K., Culver, T., Keyser, J., Lin, M., Manocha, D., 1999. Fast computation of generalized Voronoi diagrams using graphics hardware. In: Proceedings of ACM SIGGRAPH, pp. 277-286
27. Kim D.-S., Cho Y., and Kim D. Euclidean Voronoi diagram of 3D balls and its computation via tracing edges. Computer-Aided Design 37 13 (2005) 1412-1424
28. Kim D.-S., Kim D., and Sugihara K. Voronoi diagram of a circle set from Voronoi diagram of a point set: II. Geometry. Computer Aided Geometric Design 18 6 (2001) 563-585
29. Lazard S., Mariano Penaranda L., and Petitjean S. Intersecting quadrics: An efficient and exact implementation. Computational Geometry: Theory and Applications 35 1-2 (2006) 74-99. http://hal.inria.fr/inria-00000380/en/
30. Levin J. A parametric algorithm for drawing pictures of solid objects composed of quadric surfaces. Communications of the ACM 19 10 (1976) 555-563
31. Meyerovitch M. Robust, generic and efficient construction of envelopes of surfaces in three-dimensional space. Proceedings of the 14th Annual European Symposium on Algorithms (ESA). LNCS vol. 4168 (2006), Springer-Verlag 792-803
32. Meyerovitch, M., 2006a. Robust, generic and efficient construction of envelopes of surfaces in three-dimensional space. M.Sc. thesis, Department of Computer Science, Tel Aviv University, Tel Aviv, Israel
33. Myers, N., 1997. A new and useful template technique: "Traits". In: Lippman, S.B. (Ed.), C++ Gems, SIGS Reference Library, vol. 5, pp. 451-458
34. Peternell M. Geometric properties of bisector surfaces. Graphical Models 62 3 (2000) 202-236
35. Ramanathan M., and Gurumoorthy B. Constructing medial axis transform of planar domains with curved boundaries. Computer-Aided Design 35 7 (2003) 619-632
36. Ramanathan M., and Gurumoorthy B. Constructing medial axis transform of extruded and revolved 3D objects with free-form boundaries. Computer-Aided Design 37 13 (2005) 1370-1387
37. Wang W., Goldman R.N., and Tu C. Enhancing Levin's method for computing quadric-surface intersections. Computer Aided Geometric Design 20 7 (2003) 401-422
38. Wein R., Fogel E., Zukerman B., and Halperin D. 2D arrangements. In: Cgal Editorial Board (Ed.), Cgal-3.3 User and Reference Manual. http://www.cgal.org/Manual/3.3/doc_html/cgal_manual/Arrangement_2/Chapter_main.html
39. Will H.-M. Fast and efficient computation of additively weighted Voronoi cells for applications in molecular biology. Algorithm Theory-SWAT '98, 6th Scandinavian Workshop on Algorithm Theory. LNCS vol. 1432 (1998), Springer-Verlag 310-321
40. Yap C.K., and Dubé T. The exact computation paradigm. In: Du D.Z., and Hwang F.K. (Eds). Computing in Euclidean Geometry. 2nd ed. Lecture Notes Series on Computing vol. 1 (1995), World Scientific, Singapore 452-492
41. Zhang Y.Y., and Wang P.S.P. Analytic comparison of thinning algorithms. International Journal of Pattern Recognition and Artificial Intelligence 7 5 (1993) 1227-1246