Abstract
We introduce a new, efficient, and complete algorithm, and its exact implementation, to compute the Voronoi diagram of lines in space. This is a major milestone towards the robust construction of the Voronoi diagram of polyhedra. As we follow the exact geometric-computation paradigm, it is guaranteed that we always compute the mathematically correct result. The algorithm is complete in the sense that it can handle all configurations, in particular all degenerate ones. The algorithm requires O(n 3 + ε) time and space, where n is the number of lines. The Voronoi diagram is represented by a data structure that permits answering point-location queries in O(log2 n) expected time. The implementation employs the Cgal packages for constructing arrangements and lower envelopes together with advanced algebraic tools.
This work has been supported in part by the Israel Science Foundation (grant no. 236/06), by the German-Israeli Foundation (grant no. 969/07), and by the Hermann Minkowski–Minerva Center for Geometry at Tel Aviv University.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Agarwal, P.K., Schwarzkopf, O., Sharir, M.: The overlay of lower envelopes and its applications. Disc. Comput. Geom. 15(1), 1–13 (1996)
Aurenhammer, F., Klein, R.: Voronoi diagrams. In: Sack, J., Urrutia, G. (eds.) Handb. Comput. Geom., ch. 5, pp. 201–290. Elsevier, Amsterdam (2000)
Austern, M.H.: Generic Programming and the STL. Addison-Wesley, Reading (1999)
Berberich, E., Hemmer, M., Kerber, M.: A generic algebraic kernel for non-linear geometric applications. Research Report 7274, INRIA (2010)
Berberich, E., Hemmer, M., Kettner, L., Schömer, E., Wolpert, N.: An exact, complete and efficient implementation for computing planar maps of quadric intersection curves. In: Mitchell, J., Rote, G., Kettner, L. (eds.) Proc. 21st Annu. ACM Symp. Comput. Geom., pp. 99–106. ACM Press, Pisa (2005)
Blum, H.: A transformation for extracting new descriptors of shape. In: WathenDunn, W. (ed.) Models for the Perception of Speech and Visual Form. MIT Press, Cambridge (1967)
Boissonnat, J.D., Delage, C.: Convex hull and Voronoi diagram of additively weighted points. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 367–378. Springer, Heidelberg (2005)
Boissonnat, J.D., Teillaud, M. (eds.): Effective Computational Geometry for Curves and Surfaces. Mathematics and Visualization. Springer, Heidelberg (2006)
Culver, T., Keyser, J., Manocha, D.: Exact computation of the medial axis of a polyhedron. Computer Aided Geometric Design 21(1), 65–98 (2004)
Devillers, O.: Improved incremental randomized Delaunay triangulation. In: Proc. 14th Annu. ACM Symp. Comput. Geom., pp. 106–115. ACM Press, New York (1998)
Devroye, L., Lemaire, C., Moreau, J.M.: Expected time analysis for Delaunay point location. Computational Geometry 29(2), 61–89 (2004)
Dupont, L., Hemmer, M., Petitjean, S., Schömer, E.: Complete, exact and efficient implementation for computing the adjacency graph of an arrangement of quadrics. In: Arge, L., Hoffmann, M., Welzl, E. (eds.) ESA 2007. LNCS, vol. 4698, pp. 633–644. Springer, Heidelberg (2007)
Edelsbrunner, H., Seidel, R.: Voronoi diagrams and arrangements. Disc. Comput. Geom. 1, 25–44 (1986)
Emiris, I.Z., Tsigaridas, E.P., Tzoumas, G.M.: The predicates for the Voronoi diagram of ellipses. In: Proc. 22nd Annu. ACM Symp. Comput. Geom., pp. 227–236. ACM Press, New York (2006)
Emiris, I.Z., Karavelas, M.I.: The predicates of the Apollonius diagram: Algorithmic analysis and implementation. Comput. Geom. Theory Appl. 33(1-2), 18–57 (2006)
Everett, H., Gillot, C., Lazard, D., Lazard, S., Pouget, M.: The Voronoi diagram of three arbitrary lines in \({\mathbb R}^3\). In: Abstracts of 25th Eur. Workshop Comput. Geom. (2009)
Everett, H., Lazard, S., Lazard, D., Din, M.S.E.: The Voronoi diagram of three lines. In: Proc. 23rd Annu. ACM Symp. Comput. Geom., pp. 255–264. ACM Press, New York (2007)
von zur Gathen, J., Gerhard, J.: Modern Computer Algebra. Cambridge University Press, Cambridge (1999)
Halperin, D., Kavraki, L.E., Latombe, J.C.: Robotics. In: Goodman, J.E., O’Rourke, J. (eds.) Handb. Disc. Comput. Geom., 2nd edn., ch. 48, pp. 1065–1093. Chapman & Hall/CRC, Boca Raton (2004)
Hanniel, I., Elber, G.: Computing the Voronoi cells of planes, spheres and cylinders in \({\mathbb R}^3\). Comput. Aided Geom. Des. 26(6), 695–710 (2009)
Haran, I., Halperin, D.: An experimental study of point location in planar arrangements in CGAL. ACM Journal of Experimental Algorithmics 13 (2008)
Frey, P.J.: : MEDIT : An interactive Mesh visualization Software. Technical Report RT-0253, INRIA (December 2001)
Karavelas, M.I.: A robust and effient implementation for the segment Voronoi diagram. In: Int. Symp. on Voronoi Diagrams in Sci. and Engineering, pp. 51–62 (2004)
Karavelas, M.I., Yvinec, M.: Dynamic additively weighted Voronoi diagrams in 2D. In: Proc. 10th Annu. Eur. Symp. Alg., pp. 586–598. Springer, London (2002)
Kim, D.S., Seo, J., Kim, D., Cho, Y., Ryu, J.: The beta-shape and beta-complex for analysis of molecular structures. In: Gavrilova, M.L. (ed.) Generalized Voronoi Diagram: A Geometry-Based Approach to Computational Intelligence. Studies in Computational Intelligence, vol. 158, pp. 47–66. Springer, Heidelberg (2008)
Koltun, V., Sharir, M.: 3-dimensional Euclidean Voronoi diagrams of lines with a fixed number of orientations. SIAM J. on Computing 32(3), 616–642 (2003)
Milenkovic, V.: Robust construction of the Voronoi diagram of a polyhedron. In: Proc. 5th Canad. Conf. Comput. Geom., pp. 473–478 (1993)
Mulmuley, K.: A fast planar partition algorithm, I. In: Proc. 29th Annu. IEEE Sympos. Found. Comput. Sci., pp. 580–589 (1988)
Myers, N.: Traits: A new and useful template technique. C++ Gems 17 (1995)
Rineau, L., Yvinec, M.: 3D surface mesh generation. In: CGAL Editorial Board CGAL User and Reference Manual (ed.), 3.5 edn. (2009)
Setter, O., Sharir, M., Halperin, D.: Constructing two-dimensional Voronoi diagrams via divide-and-conquer of envelopes in space. Transactions on Computational Sciences (to appear, 2010)
Sharir, M.: Almost tight upper bounds for lower envelopes in higher dimensions. Disc. Comput. Geom. 12(1), 327–345 (1994)
The CGAL Project: CGAL User and Reference Manual. CGAL Editorial Board, 3.6 edn. (2010), http://www.cgal.org/
Wein, R., van den Berg, J.P., Halperin, D.: The visibility-Voronoi complex and its applications. Computational Geometry: Theory and Applications 36(1), 66–87 (2007); special Issue on the 21st European Workshop on Computational Geometry - EWCG 2005
Yaffe, E., Halperin, D.: Approximating the pathway axis and the persistence diagram of a collection of balls in 3-space. In: Proc. 24th Annu. ACM Symp. Comput. Geom., pp. 260–269. ACM Press, New York (2008)
Yap, C.K., Dubé, T.: The exact computation paradigm. In: Du, D.Z., Hwang, F.K. (eds.) Computing in Euclidean Geometry, 2nd edn. LNCS, vol. 1, pp. 452–492. World Scientific, Singapore (1995)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hemmer, M., Setter, O., Halperin, D. (2010). Constructing the Exact Voronoi Diagram of Arbitrary Lines in Three-Dimensional Space. In: de Berg, M., Meyer, U. (eds) Algorithms – ESA 2010. ESA 2010. Lecture Notes in Computer Science, vol 6346. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15775-2_34
Download citation
DOI: https://doi.org/10.1007/978-3-642-15775-2_34
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15774-5
Online ISBN: 978-3-642-15775-2
eBook Packages: Computer ScienceComputer Science (R0)