Abstract
This paper offers combinatorial results on extremum problems concerning the number of tetrahedra in a tetrahedrization of n points in general position in three dimensions, i.e. such that no four points are coplanar. It also presents an algorithm that in O(nlog n) time constructs a tetrahedrization of a set of n points consisting of at most 3nā11 tetrahedra.
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Research of the first author is supported by Amoco Fnd. Fac. Dev. Comput. Sci. 1-6-44862, the second author is supported by NSF Grant ECS 84-10902, and research of the third author is supported in part by ONR Grant N00014-85K0570 and by NSF Grant DMS 8504322.
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References
Aho, A. V., Hopcroft, J. E. and Ullman, J. D. The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading, Mass., 1974.
Avis, D. and ElGindy, H. Triangulating point sets in space. Discrete Comput. Geom. 2 (1987), 99ā111.
Bronsted, A. An Introduction to Convez Polytopes. Grad. Texts in Math., Springer-Verlag, New York, 1983.
Cavendish, J. C., Field, D. A. and Frey, W. H. An approach to automatic three-dimensional finite element mesh generation. Internat. J. Numer. Methods Engrg. 21 (1985), 329ā347.
Edelsbrunner, H. Edge-skeletons in arrangements with applications. Algorithmica 1 (1986), 93ā109.
Edelsbrunner, H., Guibas, L. J. and Stolfi, J. Optimal point location in monotone subdivisions. SIAM J. Comput. 15 (1986), 317ā340.
Edelsbrunner, H. and Seidel, R. Voronoi diagrams and arrangements. Discrete Comput. Geom. 1 (1986), 25ā44.
Greenberg, M. J. Lectures on Algebraic Topology. W. A. Benjamin, Inc., Reading, Mass., 1967.
Hopf, H. Ćber ZusammenhƤnge zwischen Topologie und Metrik im Rahmen der elementaren Geometrie. Mathematisch-Physikalische Semester Berichte 3 (1953), 16ā29.
Klee, V. On the complexity of d-dimensional Voronoi diagrams. Archiv Math. (Basel) 34 (1980), 75ā80.
Lee, D. T. and Yang, C. C. Location of multiple points in a planar subdivision. Inform. Process. Lett. 9 (1979), 190ā193.
Preparata, F. P. A note on locating a set of points in a planar subdivision. SIAM J. Comput 8 (1979), 542ā545.
Preparata, F. P. and Hong, S. J. Convex hulls of finite sets of points in two and three dimensions. Comm. ACM 20 (1977), 87ā93.
Preparata, F. P. and Shamos, M. I. Computational Geometry. Springer-Verlag, New York, 1985.
Rothschild, G. L. and Straus, E. G. On triangulations of the convex hull of n points. Combinatorica 5 (1985), 167ā179.
Seidel, R. A convex hull algorithm optimal for point sets in even dimensions. Rep.81-14, Dept. Comput. Sci., Univ. British Columbia, Vancouver, BC, 1981.
Seidel, R. The complexity of Voronoi diagrams in higher dimensions. In āProc. 20th Ann. Allerton Conf. Commun., Control, Comput. 1982ā, 94ā95.
Sleator, D. D., Tarjan, R. E. and Thurston, W. P. Rotation distance, triangulations, and hyperbolic geometry. In āProc. 18th Ann. ACM Sympos. Theory Comput. 1986ā, 122ā135.
Strang, G. and Fix, G. An Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs, NJ, 1973.
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Edelsbrunner, H., Preparata, F.P., West, D.B. (1989). Tetrahedrizing point sets in three dimensions. In: Gianni, P. (eds) Symbolic and Algebraic Computation. ISSAC 1988. Lecture Notes in Computer Science, vol 358. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-51084-2_31
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DOI: https://doi.org/10.1007/3-540-51084-2_31
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