Abstract
Two typical approaches to robust geometric algorithms are applied to the construction of the three-dimensional convex hull, and their performances are measured by experiments. One of them is the topology-oriented approach and the other is the exact arithmetic approach accompanied with the symbolic perturbation and the floating-point acceleration. The merits and demerits of the two approaches were investigated by computational experiments.
The work was partly supported by the Grant-in-Aid for Scientific Research of the Ministry of Education, Science and Culture of Japan.
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References
M. Benouamer, D. Michelucci and B. Peroche. Error-free boundary evaluation using lazy rational arithmetic — A detailed implementation. Proc. of the 2nd Symp. on Solid Modeling and Appl., Montreal, 115–126, 1993.
A. M. Day. The implementation of an algorithm to find the convex hull of a set of three-dimensional points. ACM Trans. on Graphics, 9: 105–132, 1990.
H. Edelsbrunner. Algorithms in Combinatorial Geometry. Springer-Verlag, Berlin, 1987.
H. Edelsbrunner and E. P. Mücke. Simulation of simplicity—A technique to cope with degenerate cases in geometric algorithms. Proc. Of the 4th ACM Symp. on Comput. Geom., Urbana-Champaign, 118–133, 1988.
S. Fortune and C. von Wyk. Efficient exact arithmetic for computational geometry. Proc. of the 9th ACM Annual Symp. on Comput. Geom., San Diego, 163–172, 1993.
L. Guibas, D. Salesin and J. Stolfi. Epsilon geometry — Building robust algorithms from imprecise calculations. Proc. of the 5th ACM Annual Symp. on Comput. Geom., Saarbrücken, 208–217, 1989.
V. Milenkovic. Verifiable implementation of geometric algorithms using finite precision arithmetic. Artif. Intell., 37: 377–401, 1988.
T. Ono. An eight-way perturbation technique for the three-dimensional convex hull. Proc. of the 6th Canadian Conf. on Comput. Geom., Saskatoon, 159–164, 1994.
F. P. Preparata and M. I. Shamos. Computational Geometry — An Introduction. Springer-Verlag, New York, 1985.
P. Schorn. Robust algorithms in a program library for geometric computation. Dissertation submitted to the Swiss Federal Institute of Technology (ETH) Zürick for the degree of Doctor of Technical Sciences, 1991.
E. Steinitz. Polyheder und Raumeinteilungen, Encykloädie der mathematischen Wissenchaften, Band III, Teil 1, 2. Hälfte, IIIAB12, 1–139, 1916.
K. Sugihara. A robust and consistent algorithm for intersecting convex polyhedra. EUROGRAPHICS'94, Oslo, C-45-C-54,1994.
K. Sugihara and M. Iri. Construction of the Voronoi diagram for “one million” generators in single-precision arithmetic. Proc. of the IEEE, 80: 1471–1488, 1992.
C. K. Yap. Symbolic treatment of geometric degeneracies. J. of Symbolic Comput., 10: 349–370, 1990.
C. K. Yap. Toward exact geometric computation. Proc. of the 5th Canadian Conf. on Comput. Geom., Waterloo, 405–419, 1993.
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© 1997 Springer-Verlag Berlin Heidelberg
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Minakawa, T., Sugihara, K. (1997). Topology oriented vs. exact arithmetic — Experience in implementing the three-dimensional convex hull algorithm. In: Leong, H.W., Imai, H., Jain, S. (eds) Algorithms and Computation. ISAAC 1997. Lecture Notes in Computer Science, vol 1350. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63890-3_30
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DOI: https://doi.org/10.1007/3-540-63890-3_30
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