Abstract
Voronoi diagrams tessellate the space where each cell corresponds to an associated generator under an a priori defined distance and have been extensively used to solve geometric problems of various disciplines. Additively-weighted Voronoi diagrams, also called the Voronoi diagram of disks and spheres, have many critical applications and a few algorithms are known. However, algorithmic robustness remains a major hurdle to use these Voronoi diagrams in practice. There are two important yet different approaches to design robust algorithms: the exact-computation and topology-oriented approaches. The former uses high-precision arithmetic and guarantees the correctness mathematically with the cost of a significant use of computational resources. The latter focuses on topological properties to keep consistency using logical computation rather than numerical computation. In this paper, we present a robust and efficient algorithm for computing the Voronoi diagram of disks using a topology-oriented incremental method. The algorithm is rather simple as it primarily checks topological changes only during each disk is incrementally inserted into a previously constructed Voronoi diagram of some other disks.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Okabe, A., Boots, B., Sugihara, K., Chiu, S.N.: Spatial Tessellations: Conceptsand Applications of Voronoi Diagrams, 2nd edn. Wiley, Chichester (1999)
Aurenhammer, F.: Voronoi diagrams - a survey of a fundamental geometric data structure. ACM Comput. Surv. 23(3), 345–405 (1991)
Kim, D.-S., Kim, D., Sugihara, K.: Voronoi diagram of a circle set from Voronoi diagram of a point set: I. topology. Comput. Aided Geom. Des. 18, 541–562 (2001)
Kim, D.-S., Kim, D., Sugihara, K.: Voronoi diagram of a circle set from Voronoi diagram of a point set: II. geometry. Comput. Aided Geom. Des. 18, 563–585 (2001)
Held, M. (ed.): On the Computational Geometry of Pocket Machining. LNCS, vol. 500. Springer, Heidelberg (1991)
Kim, D.-S., Hwang, I.-K., Park, B.-J.: Representing the Voronoi diagram of a simple polygon using rational quadratic Bézier curves. Comput. Aided Des. 27(8), 605–614 (1995)
Kim, D.-S., Ryu, J., Shin, H., Cho, Y.: Beta-decomposition for the volume and area of the union of three-dimensional balls and their offsets. J. Comput. Chem. 33(13), 1252–1273 (2012)
Kim, J.-K., Cho, Y., Kim, D., Kim, D.-S.: Voronoi diagrams, quasi-triangulations, and beta-complexes for disks in \(\mathbb{R}^2\): The theory and implementation in BetaConcept. J. Comput. Des. Eng. 1(2), 79–87 (2014)
Lee, D., Drysdale, R.: Generalization of Voronoi diagrams in the plane. SIAM J. Comput. 10(1), 73–87 (1981)
Sharir, M.: Intersection and closest-pair problems for a set of planar discs. SIAM J. Comput. 14(2), 448–468 (1985)
Yap, C.-K.: An \({O}(n \log n)\) algorithm for the Voronoi diagram of a set of simple curve segments. Discrete Comput. Geom. 2, 365–393 (1987)
Fortune, S.: A sweepline algorithm for Voronoi diagrams. Algorithmica 2, 153–174 (1987)
Sugihara, K.: Approximation of generalized Voronoi diagrams by ordinary Voronoi diagrams. Graphical Models Image Process. 55(6), 522–531 (1993)
Karavelas, M., Emiris, I.Z.: Predicates for the planar additively weighted Voronoi diagram, Technical Report ECG-TR-122201-01. INRIA Sophia-Antipolis, Sophia-Antipolis (2002)
Sugihara, K., Iri, M.: A solid modelling system free from topological lnconsistency. J. Inf. Process. 12(4), 380–393 (1989)
Sugihara, K.: A simple method for avoiding numerical errors and degeneracy in Voronoi diagram construction. IEICE Trans. Fundam. E75–A, 468–477 (1992)
Yap, C.-K.: Towards exact geometric computation. Comput. Geom. Theory Appl. 7(1–2), 3–23 (1997)
Yu, J., Zhou, Y., Tanaka, I., Yao, M.: Roll: a new algorithm for the detection of protein pockets and cavities with a rolling probe sphere. Struct. Bioinf. 26(1), 46–52 (2010)
Fabri, A., Giezeman, G.J., Kettner, L., Schirra, S., Schönherr, S.: On the design of CGAL a computational geometry algorithms library. Softw. Pract. Experience 30(11), 1167–1202 (2000)
Goodman, J.E., ORourke, J.: Handbook of Discrete and Computational Geometry. CRC Press, Boca Raton (1997)
Sugihara, K., Iri, M.: Construction of the Voronoi diagram for “one million” generators in single-precision arithmetic. Proc. IEEE 80(9), 1471–1484 (1992)
Sugihara, K., Iri, M.: A robust topology-oriented incremental algorithm for Voronoi diagrams. Int. J. Comput. Geom. Appl. 4(2), 179–228 (1994)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Lee, M., Sugihara, K., Kim, DS. (2016). Robust Construction of the Additively-Weighted Voronoi Diagram via Topology-Oriented Incremental Algorithm. In: Greuel, GM., Koch, T., Paule, P., Sommese, A. (eds) Mathematical Software – ICMS 2016. ICMS 2016. Lecture Notes in Computer Science(), vol 9725. Springer, Cham. https://doi.org/10.1007/978-3-319-42432-3_66
Download citation
DOI: https://doi.org/10.1007/978-3-319-42432-3_66
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-42431-6
Online ISBN: 978-3-319-42432-3
eBook Packages: Computer ScienceComputer Science (R0)