Abstract
This paper is concerned with the problem of partitioning a three-dimensional nonconvex polytope into a small number of elementary convex parts. The need for such decompositions arises in tool design, computer-aided manufacturing, finite-element methods, and robotics. Our main result is an algorithm for decomposing a nonconvex polytope of zero genus withn vertices andr reflex edges intoO(n +r 2) tetrahedra. This bound is asymptotically tight in the worst case. The algorithm requiresO(n +r 2) space and runs inO((n +r 2) logr) time.
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References
B. Aronov and M. Sharir, Triangles in Space, or Building and Analyzing Castles in the Air,Proc. 4th Ann. ACM Symp. Comput. Geom. (1988), 381–391.
C. L. Bajaj and T. K. Dey, Robust Decompositions of Polyhedra, Dept. Computer Science, Purdue University, 1989.
T. J. Baker, Three-Dimensional Mesh Generation by Triangulation of Arbitrary Point Sets, Dept. Mechanical Engineering, Princeton University, 1986.
B. S. Baker, E. Grosse, and C. S. Rafferty, Non-Obtuse Triangulation of Polygons,Discrete Comput. Geom. 3 (1988), 147–168.
B. G. Baumgart, A Polyhedron Representation for Computer Vision,Proc. 1975 National Comput. Conf., saAFIPS Conference Proceedings, Vol. 44, AFIPS Press, Montvale, NJ, 1975, 589–596.
J. F. Canny, A new Algebraic Method for Motion Planning and Real Geometry,Proc. 28th Ann. IEEE Symp. on Found. Comput. Sci. (1987), 39–48.
B. Chazelle, Convex Partitions of Polyhedra: A Lower Bound and Worst-Case Optimal Algorithm,SIAM J. Comput. 13 (1984), 488–507.
B. Chazelle, Approximation and Decomposition of Shapes,Advances in Robotics, Vol. 1 (J. T. Schwartz and C. K. Yap, ed.), Erlbaum, Hillsdale, NJ, 1987, 145–185.
B. Chazelle, H. Edelsbrunner, L. J. Guibas, and M. Sharir, A Singly-Exponential Stratification Scheme for Real Semi-Algebraic Varieties and Its Applications,Proc. 16th ICALP, Lecture Notes in Computer Science, Vol. 372, Springer-Verlag, Berlin, 1989, 179–193.
B. Chazelle, L. J. Guibas, and D. T. Lee, The Power of Geometric Duality,BIT 25 (1985), 76–90.
G. E. Collins, Quantifier Elimination for Real Closed Fields by Cylindric Algebraic Decomposition,Proc. 2nd Gl Conf. Automata Theory and Formal Languages, Lecture Notes in Computer Science, Vol 33, Springer-Verlag, Berlin, 1975, 134–183.
D. P. Dobkin and M. J. Laszlo, Primitives for the Manipulation of Three-Dimensional Subdivisions,Proc. 3rd Ann. ACM Symp. Comput. Geom. (1987), 86–99.
H. Edelsbrunner, L. J. Guibas, and J. Stolfi, Optimal Point Location in a Monotone Subdivision,SIAM J. Comput. 15 (1986), 317–340.
H. Edelsbrunner, J. O'Rourke, and R. Seidel, Constructing Arrangements of Lines and Hyperplanes with Applications,SIAM J. Comput. 15 (1986), 341–363.
H. Feng and T. Pavlidis, Decomposition of Polygons into Simpler Components: Feature Generation for Syntactic Pattern Recognition,IEEE Trans. Comput. 24 (1975), 636–650.
D. A. Field, Implementing Watson's Algorithm in Three Dimensions,Proc. 2nd Ann. ACM Symp. Comput. Geom. (1986), 246–259.
L. J. Guibas and J. Stolfi, Primitives for the Manipulating of General Subdivisions and the Computation of Voronoi Diagrams,ACM Trans. Graphics 4 (1985), 75–123.
D. G. Kirkpatrick Optimal Search in Planar Subdivisions,SIAM J. Comput. 12 (1983), 28–35.
A. Lingas, The Power of Non-Rectilinear Holes,Proc. 9th Colloq. Automata, Languages and Programming, Lecture Notes in Computer Science, Vol. 140, Springer-Verlag, Berlin, 1982, 369–383.
K. Mehlhorn,Data Structures and Algorithms, Vol. 3, Springer-Verlag, Berlin, 1984.
D. E. Muller and F. P. Preparata, Finding the Intersection of Two Convex Polyhedra,Theoret. Comput. Sci. 7 (1978), 217–236.
J. O'Rourke,Art Gallery Theorems and Algorithms, Oxford University Press, Oxford, 1987.
F. P. Preparata and M. I. Shamos,Computational Geometry: An Introduction, Springer-Verlag, New York, 1985.
D. Prill, On Approximations and Incidence in Cylindrical Algebraic Decompositions,SIAM J. Comput. 15 (1986), 972–993.
J. Ruppert and R. Seidel, On the Difficulty of Tetrahedralizing 3-Dimensional Non-Convex Polyhedra,Proc. 5th Ann. ACM Symp Comput. Geom. (1989), 380–392.
B. Schachter, Decomposition of Polygons into Convex Sets,IEEE Trans. Comput. 27 (1978), 1078–1082.
J. T. Schwartz and M. Sharir, On the “Piano Movers” Problem, II: General Techniques for Computing Topological Properties of Real Algebraic Manifolds,Adv. in Appl. Math. 4 (1983), 298–351.
W. Smith, Studies in Computational Geometry Motivated by Mesh Generation, Ph.D. Thesis, Princeton University, 1988.
H. Whitney, Elementary Structure of Real Algebraic Varieties,Ann. of Math. 66 (1957), 545–556.
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This research was supported in part by the National Science Foundation under Grant CCR-8700917.
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Chazelle, B., Palios, L. Triangulating a nonconvex polytope. Discrete Comput Geom 5, 505–526 (1990). https://doi.org/10.1007/BF02187807
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DOI: https://doi.org/10.1007/BF02187807