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Computing the width of a set
September 1988 (vol. 10 no. 5)
pp. 761,762,763,764,765
For a set of points P in three-dimensional space, the width of P, W (P), is defined as the minimum distance between parallel planes of support of P. It is shown that W(P) can be computed in O(n log n+I) time and O(n) space, where I is the number of antipodal pairs of edges of the convex hull of P, and n is the number of vertices; in the worst case, I=O(n/sup 2/). For a convex polyhedra the time complexity becomes O(n+I). If P is a set of points in the plane, the complexity can be reduced to O(nlog n). For simple polygons, linear time suffices.<>

[1] G. T. Toussaint, "Movable separability of sets," inComputational Geometry, G. T. Toussaint, Ed. Amsterdam, The Netherlands: North-Holland, 1985, pp. 335-375.
[2] M. I. Shamos, "Computational geometry," Ph.D. dissertation, Dep. Comput. Sci., Yale Univ., 1978.
[3] K. Q. Brown, "Geometric transforms for fast geometric algorithms," Dep. Comput. Sci., Carnegie-Mellon Univ., 1979.
[4] D. G. Kirkpatrick and R. Seidel, "The ultimate planar convex hull algorithm?", Dep. Comput. Sci., Cornell Univ., Tech. Rep. 83-577, Oct. 1982.
[5] M. M. McQueen and G. T. Toussaint, "On the ultimate convex hull algorithm in practice,"Pattern Recognition Lett., pp. 29-34, Jan. 1985.
[6] D. McCallum and D. Avis, "A linear time algorithm for finding the convex hull of a simple polygon,"Inform. Processing Lett., pp. 201- 205, Dec. 1979.
[7] D. T. Lee, "On finding the convex hull of a simple polygon," Tech. Northwestern Univ., Tech. Rep. 80-03-FC-01, Mar. 1980.
[8] L. Guibas and R. Seidel, "Computing convolutions by reciprocal search," inProc. 2nd Symp. Comput. Geom., Yorktown Heights, NY, 1986, pp. 90-99.
[9] M. E. Houle and G. T. Toussaint, "Computmng the width of a set," Tech. Rep. SOCS-84.22, Dec. 1984.
[10] F. P. Preparata and S. J. Hong, "Convex hulls of finite sets of points in two and three dimensions,"Commun. ACM, vol. 20, pp. 87-93, 1977.
[11] K. Ichida and T. Kiyono, "Segmentation of plane curves,"Trans. Elec. Commun. Eng., Japan, vol. 58-D, pp. 689-696, 1975.
[12] H. Imai and M. Iri, "Polygonal approximations of a curve: Formulations and solution algorithms," inComputational Morphology, G. T. Toussaint, Ed. Amsterdam, The Netherlands: North Holland, to be published.
[13] Y. Kurozumi and W. A. Davis, "Polygonal approximation by the minimax method,"Comput. Graphics Image Processing, vol. 19, pp. 248-264, 1982.
[14] G. Toussaint, "Solving geometric problems with the rotating calipers," inProc. MELECON '83, Athens, Greece, 1983.
[15] G. Toussaint, "Approximating polygonal curves in three-dimensions," manuscript in preparation.

Index Terms:
set theory,computational complexity,computational geometry,pattern recognition,polygons,3D space,point sets,computational complexity,pattern recognition,computational geometry,parallel planes,convex hull,time complexity,Concurrent computing,Artificial intelligence,Computational geometry,Image processing,Minimax techniques,Pattern recognition,Canada Councils,Computer science,Terminology,Euclidean distance
"Computing the width of a set," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 10, no. 5, pp. 761,762,763,764,765, Sept. 1988, doi:10.1109/34.6790
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