Abstract
We show that any locally-fat (or α, β-covered) polyhedron with convex fat faces can be decomposed into O(n) tetrahedra, where n is the number of vertices of the polyhedron. We also show that the restriction that the faces are fat is necessary: there are locally-fat polyhedra with non-fat faces that require Ω(n 2) pieces in any convex decomposition. Furthermore, we show that if we want the polyhedra in the decomposition to be fat themselves, then the worst-case number of tetrahedra cannot be bounded as a function of n. Finally, we obtain several results on the problem where we want to only cover the boundary of the polyhedron, and not its entire interior.
This research was supported by the Netherlands’ Organisation for Scientific Research (NWO) under project no. 639.023.301.
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References
Aronov, B., de Berg, M., Gray, C.: Ray shooting and intersection searching amidst fat convex polyhedra in 3-space. Computational Geometry: Theory and Applications 41, 68–76 (2008)
Aronov, B., Sharir, M.: On translational motion planning of a convex polyhedron in 3-space. SIAM J. Comput. 26, 1785–1803 (1997)
de Berg, M.: Ray Shooting, Depth Orders and Hidden Surface Removal. LNCS, vol. 703. Springer, New York (1993)
de Berg, M.: Linear size binary space partitions for uncluttered scenes. Algorithmica 28, 353–366 (2000)
de Berg, M.: Improved bounds on the union complexity of fat objects. Discr. Comput. Geom. (to appear) doi:10.1007/s00454-007-9029-7
de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer, Heidelberg (2008)
de Berg, M., Gray, C.: Vertical ray shooting and computing depth orders for fat objects. SIAM J. Comput. 38(1), 257–275 (2008)
de Berg, M., Gray, C.: Computing the visibility map of fat objects. In: Dehne, F., Sack, J.-R., Zeh, N. (eds.) WADS 2007. LNCS, vol. 4619, pp. 251–262. Springer, Heidelberg (2007)
de Berg, M., David, H., Katz, M.J., Overmars, M., van der Stappen, A.F., Vleugels, J.: Guarding scenes against invasive hypercubes. Comput. Geom. Theory Appl. 26, 99–117 (2003)
de Berg, M., van der Stappen, A.F., Vleugels, J., Katz, M.J.: Realistic input models for geometric algorithms. Algorithmica 34, 81–97 (2002)
Chazelle, B.: Convex partitions of polyhedra: a lower bound and worst-case optimal algorithm. SIAM J. Comput. 13, 488–507 (1984)
Chazelle, B.: Triangulating a simple polygon in linear time. Discr. Comput. Geom. 6, 485–524 (1991)
Efrat, A.: The complexity of the union of (α, β)-covered objects. SIAM J. Comput. 34, 775–787 (2005)
Erickson, J.: Local polyhedra and geometric graphs. Comput. Geom. Theory Appl. 31, 101–125 (2005)
Hachenberger, P.: Exact Minkowski sums of polyhedra and exact and efficient decomposition of polyhedra in convex pieces. In: Arge, L., Hoffmann, M., Welzl, E. (eds.) ESA 2007. LNCS, vol. 4698, pp. 669–680. Springer, Heidelberg (2007)
Keil, J.M.: Polygon Decomposition. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 491–518 (2000)
van Kreveld, M.: On fat partitioning, fat covering, and the union size of polygons. Comput. Geom. Theory Appl. 9, 197–210 (1998)
Rupert, J., Seidel, R.: On the difficulty of triangulating three-dimensional nonconvex polyhedra. Discr. Comput. Geom. 7, 227–253 (1992)
van der Stappen, A.F.: Motion planning amidst fat obstacles. Ph.D. thesis, Utrecht University, Utrecht, the Netherlands (1994)
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de Berg, M., Gray, C. (2008). Decompositions and Boundary Coverings of Non-convex Fat Polyhedra. In: Halperin, D., Mehlhorn, K. (eds) Algorithms - ESA 2008. ESA 2008. Lecture Notes in Computer Science, vol 5193. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87744-8_15
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DOI: https://doi.org/10.1007/978-3-540-87744-8_15
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