An optimal algorithm for intersecting three-dimensional convex polyhedra

A linear algorithm for intersecting two convex polyhedra in 3-space is described. The algorithm is quite simple; it does not require any complicated data structure and should be practical. A number of optimal algorithms for other problems are obtained directly from this result. These include intersecting several polytopes at once or computing the convex hull of their union, merging Voronoi diagrams in the plane in linear time, and computing three-dimensional convex hulls in linear expected time.<<ETX>>

[1]  Nimrod Megiddo,et al.  Linear-time algorithms for linear programming in R3 and related problems , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[2]  David P. Dobkin,et al.  Efficient uses of the past , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[3]  F. P. Preparata,et al.  Convex hulls of finite sets of points in two and three dimensions , 1977, CACM.

[4]  Michael Ian Shamos,et al.  Geometric intersection problems , 1976, 17th Annual Symposium on Foundations of Computer Science (sfcs 1976).

[5]  Chee-Keng Yap.  Geometric Consistency Theorem for a Symbolic Perturbation Scheme , 1990, J. Comput. Syst. Sci..

[6]  David G. Kirkpatrick,et al.  Optimal Search in Planar Subdivisions , 1983, SIAM J. Comput..

[7]  Michael Ian Shamos,et al.  Divide and Conquer for Linear Expected Time , 1978, Inf. Process. Lett..

[8]  Bernard Chazelle,et al.  Intersection of convex objects in two and three dimensions , 1987, JACM.

[9]  Leonidas J. Guibas,et al.  A linear-time algorithm for computing the voronoi diagram of a convex polygon , 1989, Discret. Comput. Geom..

[10]  David E. Muller,et al.  Finding the Intersection of two Convex Polyhedra , 1978, Theor. Comput. Sci..

[11]  Bruce G. Baumgart.  A polyhedron representation for computer vision , 1975, AFIPS '75.

[12]  JOSEPH O’ROURKE,et al.  A new linear algorithm for intersecting convex polygons , 1982, Comput. Graph. Image Process..

[13]  Kurt Mehlhorn,et al.  Data Structures and Algorithms 3: Multi-dimensional Searching and Computational Geometry , 2012, EATCS Monographs on Theoretical Computer Science.

[14]  Leonidas J. Guibas,et al.  Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams , 1983, STOC.

[15]  David G. Kirkpatrick,et al.  Efficient computation of continuous skeletons , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[16]  Raimund Seidel,et al.  Voronoi diagrams and arrangements , 1986, Discret. Comput. Geom..

[17]  David G. Kirkpatrick,et al.  Fast Detection of Polyhedral Intersection , 1983, Theor. Comput. Sci..

[18]  Herbert Edelsbrunner,et al.  Finding Extreme Points in Three Dimensions and Solving the Post-Office Problem in the Plane , 1985, Inf. Process. Lett..

[19]  Herbert Edelsbrunner,et al.  Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms , 1988, SCG '88.

[20]  Herbert Edelsbrunner,et al.  Algorithms in Combinatorial Geometry , 1987, EATCS Monographs in Theoretical Computer Science.

[21]  David G. Kirkpatrick,et al.  A Linear Algorithm for Determining the Separation of Convex Polyhedra , 1985, J. Algorithms.

[22]  Micha Sharir,et al.  Triangles in space or building (and analyzing) castles in the air , 1990, Comb..

[23]  Martin E. Dyer,et al.  Linear Time Algorithms for Two- and Three-Variable Linear Programs , 1984, SIAM J. Comput..

[24]  Kurt Mehlhorn,et al.  Intersecting two polyhedra one of which is convex , 1985, FCT.

[25]  C. Rourke,et al.  Introduction to Piecewise-Linear Topology , 1972 .