Abstract
We show that the boundary of a three-dimensional polyhedron withr reflex angles and arbitrary genus can be subdivided intoO(r) connected pieces, each of which lies on the boundary of its convex hull. A remarkable feature of this result is that the number of these convex-like pieces is independent of the number of vertices. Furthermore, it is linear inr, which contrasts with a quadratic worst-case lower bound in the number of convex pieces needed to decompose the polyhedron itself. The number of new vertices introduced in the process isO(n). The decomposition can be computed inO(n+rlogr) time.
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References
E. Arkin, R. Connelly, and J. S. B. Mitchell, On Monotone Paths Among Obstacles, with Applications to Planning Assemblies,Proc. 5th Annual ACM Symposium on Computational Geometry (1989) pp. 334–343.
C. L. Bajaj and T. K. Dey, Convex Decompositions of Polyhedra and Robustness,SIAM Journal on Computing 21 (1992), 339–364.
B. G. Baumgart, A Polyhedron Representation for Computer Vision,Proc. 1975National Computer Conference, AFIPS Conference Proceedings, Vol. 44 (1975), AFIPS Press, Montvale, NJ, pp. 589–596.
B. Chazelle, Convex Partitions of Polyhedra: A Lower Bound and Worst Cast Optimal Algorithm,SIAM Journal on Computing 13 (1984), 488–507.
B. Chazelle and L. J. Guibas, Visibility and Intersection Problems in Plane Geometry,Discrete and Computational Geometry 4 (1989), 551–581.
B. Chazelle and L. Palios, Triangulating a Nonconvex Polytope,Discrete and Computational Geometry 5 (1990), 505–526.
L. J. Guibas, J. Hershberger, D. Leven, M. Sharir, and R. E. Tarjan, Linear Time Algorithms for Visibility and Shortest Path Problems Inside Triangulated Simple Polygons,Algorithmica 2 (1987), 209–233.
L. J. Guibas and J. Stolfi, Primitives for the Manipulation of General Subdivisions and the Computation of Voronoi Diagrams,ACM Transactions on Graphics 4 (1985), 75–123.
S. Hertel and K. Mehlhorn,Fast Triangulation of a Simple Polygon, Lecture Notes on Computer Science, Vol. 158 (1983), Springer-Verlag, Berlin, pp. 207–218.
D. G. Kirkpatrick, Optimal Search in Planar Subdivisions,SIAM Journal on Computing 12 (1983), 28–35.
A. Lingas,The Power of Non-Rectilinear Holes, Lecture Notes in Computer Science, Vol. 140 (1982), Springer-Verlag, Berlin, pp. 369–383.
D. E. Muller and F. P. Preparata, Finding the Intersection of two Convex Polyhedra,Theoretical Computer Science 7 (1978), 217–236.
J. O'Rourke,Art Gallery Theorems and Algorithms, Oxford University Press, Oxford (1987).
J. Ruppert and R. Seidel, On the Difficulty of Tetrahedralizing 3-Dimensional Non-Convex Polyhedra,Proc. 5th Annual ACM Symposium on Computational Geometry (1989), pp. 380–392.
S. Suri, A Linear Time Algorithm for Minimum Link Paths Inside a Simple Polygon,Computer Vision, Graphics, and Image Processing 35 (1986), 99–110.
G. T. Toussaint, On Separating Two Simple Polygons by a Single Translation,Discrete and Computational Geometry 4 (1989), 265–278.
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Communicated by L. J. Guibas.
This work was mostly done while the author was a graduate student at Princeton University.
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Chazelle, B., Palios, L. Decomposing the boundary of a nonconvex polyhedron. Algorithmica 17, 245–265 (1997). https://doi.org/10.1007/BF02523191
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DOI: https://doi.org/10.1007/BF02523191