Abstract
The problem of computing a d-dimensional Euclidean Voronoi diagram of spheres is relevant to many areas, including computer simulation, motion planning, CAD, and computer graphics. This paper presents a new algorithm based on the explicit computation of the coordinates and radii of Euclidean Voronoi diagram vertices for a set of spheres. The algorithm is further applied to compute the Voronoi diagram with a specified precision in a fixed length floating-point arithmetic. The algorithm is implemented using the ECLibrary (Exact Computation Library) and tested on the example of a 3-dimensional Voronoi diagram of a set of spheres.
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
Blum, L., Cucker, F., Shub, M. and Smale, S. “Complexity and Real Computation” (1997)
Dey, T.K., Sugihara K. and Bajaj, C. L. DT in three dimensions with finite precision arithmetic, Comp. Aid. Geom. Des 9 (1992) 457–470
Edelsbrunner, H. and Mcke, E. Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms, 4th Annual ACM Symposium on Computational Geometry (1988) 118–133.
Fortune, S. and Wyk, C. Efficient exact arithmetic for computational geometry, 4th Annual ACM Symp. Comput. Geometry, (1993) 163–172
Gavrilova, M. and Rokne, J. Reliable line segment intersection testing, Computer Aided Design, (2000) 32, 737–745.
Gavrilova, M., Ratschek, H. and Rokne, J. Exact computation of Voronoi diagram and Delaunay triangulation, Reliable Computing, (2000) 6(1) 39–60.
S. Krishnan, M. Foskey, T. Culver, J. Keyser, D. Manocha, PRECISE: Efficient Multiprecision Evaluation of Algebraic Roots and Predicates for Reliable Geometric Computations, Symp. on Computat. Geometry (2002)
Naher, S. The LEDA user manual, Version 3.1 (Jan. 16, 1995). Available from http://ftp.mpi-sb.mpg.de in directory /pub/LEDA.
Ratschek, H. and Rokne, J. Exact computation of the sign of a finite sum, Applied Mathematics and Computation, (1999) 99, 99–127.
Rokne, J.: Interval arithmetic. In: Graphics Gems III, Academic Press, pp. 61–66 and pp. 454–457 (1992).
Sugihara, K. and Iri, M. A robust topology-oriented incremental algorithm for Voronoi diagrams, IJCGA, (1994) 4 (2): 179–228.
Yap, C., Dub, T. The exact computation paradigm, In “Computing in Euclidean Geometry” (2nd Edition). Eds. D.-Z. Du and F.K. Hwang, World Scientific Press (1995)
Gavrilova, M. (2002) Algorithm library development for complex biological and mechanical systems: functionality, interoperability and numerical stability DIMACS Workshop on Implementation of Geometric Algorithms, December 2002 (abstract)
Gavrilova, M. (2002) A Reliable Algorithm for Computing the Generalized Voronoi Diagram for a Set of Spheres in the Euclidean d-dimensional Space, in the Proceedings of the 14th Canadian Conference on Computational Geometry, August, Lethbridge, Canada
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gavrilova, M.L. (2003). An Explicit Solution for Computing the Euclidean d-dimensional Voronoi Diagram of Spheres in a Floating-Point Arithmetic. In: Kumar, V., Gavrilova, M.L., Tan, C.J.K., L’Ecuyer, P. (eds) Computational Science and Its Applications — ICCSA 2003. ICCSA 2003. Lecture Notes in Computer Science, vol 2669. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44842-X_84
Download citation
DOI: https://doi.org/10.1007/3-540-44842-X_84
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40156-8
Online ISBN: 978-3-540-44842-6
eBook Packages: Springer Book Archive