The quickhull algorithm for convex hulls

The convex hull of a set of points is the smallest convex set that contains the points. This article presents a practical convex hull algorithm that combines the two-dimensional Quickhull algorithm with the general-dimension Beneath-Beyond Algorithm. It is similar to the randomized, incremental algorithms for convex hull and delaunay triangulation. We provide empirical evidence that the algorithm runs faster when the input contains nonextreme points and that it used less memory. computational geometry algorithms have traditionally assumed that input sets are well behaved. When an algorithm is implemented with floating-point arithmetic, this assumption can lead to serous errors. We briefly describe a solution to this problem when computing the convex hull in two, three, or four dimensions. The output is a set of “thick” facets that contain all possible exact convex hulls of the input. A variation is effective in five or more dimensions.

[1]  H. Raiffa,et al.  3. The Double Description Method , 1953 .

[2]  B. Grünbaum.  Measures of symmetry for convex sets , 1963 .

[3]  V. Klee.  CONVEX POLYTOPES AND LINEAR PROGRAMMING. , 1964 .

[4]  Donald R. Chand,et al.  An Algorithm for Convex Polytopes , 1970, JACM.

[5]  William F. Eddy,et al.  A New Convex Hull Algorithm for Planar Sets , 1977, TOMS.

[6]  A. Bykat,et al.  Convex Hull of a Finite Set of Points in Two Dimensions , 1978, Inf. Process. Lett..

[7]  Kevin Q. Brown,et al.  Voronoi Diagrams from Convex Hulls , 1979, Inf. Process. Lett..

[8]  Bernard W. Silverman,et al.  Constructing the Convex Hull of a Set of Points in the Plane , 1979, Comput. J..

[9]  Michael Ian Shamos,et al.  Computational geometry: an introduction , 1985 .

[10]  David G. Kirkpatrick,et al.  The Ultimate Planar Convex Hull Algorithm? , 1986, SIAM J. Comput..

[11]  Raimund Seidel,et al.  Constructing higher-dimensional convex hulls at logarithmic cost per face , 1986, STOC '86.

[12]  Kenneth L. Clarkson,et al.  Applications of random sampling in computational geometry, II , 1988, SCG '88.

[13]  Steven Fortune,et al.  Stable maintenance of point set triangulations in two dimensions , 1989, 30th Annual Symposium on Foundations of Computer Science.

[14]  Zhenyu Li,et al.  Constructing strongly convex hulls using exact or rounded arithmetic , 1990, SCG '90.

[15]  F. Frances Yao,et al.  Computational Geometry , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[16]  Franz Aurenhammer,et al.  Voronoi diagrams—a survey of a fundamental geometric data structure , 1991, CSUR.

[17]  Barry Joe,et al.  Construction of three-dimensional Delaunay triangulations using local transformations , 1991, Comput. Aided Geom. Des..

[18]  Kokichi Sugihara.  Topologically Consistent Algorithms Realted to Convex Polyhedra , 1992, ISAAC.

[19]  Kokichi Sugihara,et al.  Delaunay triangulations in three dimensions with finite precision arithmetic , 1992, Comput. Aided Geom. Des..

[20]  Incremental topological flipping works for regular triangulations , 1992, SCG '92.

[21]  Kenneth L. Clarkson,et al.  Safe and effective determinant evaluation , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[22]  J. Weeks.  Convex hulls and isometries of cusped hyperbolic 3-manifolds , 1993 .

[23]  Kurt Mehlhorn,et al.  Four Results on Randomized Incremental Constructions , 1992, Comput. Geom..

[24]  Jean-Daniel Boissonnat,et al.  On the Randomized Construction of the Delaunay Tree , 1993, Theor. Comput. Sci..

[25]  J. Boardman.  Automating spectral unmixing of AVIRIS data using convex geometry concepts , 1993 .

[26]  Ketan Mulmuley,et al.  Computational geometry : an introduction through randomized algorithms , 1993 .

[27]  Bo Zhang,et al.  Invariant sets for general second-order low-pass delta-sigma modulators with DC inputs , 1994, Proceedings of IEEE International Symposium on Circuits and Systems - ISCAS '94.

[28]  David Avis,et al.  How good are convex hull algorithms? , 1995, SCG '95.

[29]  Komei Fukuda,et al.  Double Description Method Revisited , 1995, Combinatorics and Computer Science.

[30]  Bernard Chazelle,et al.  Derandomizing an Output-sensitive Convex Hull Algorithm in Three Dimensions , 1995, Comput. Geom..

[31]  Wayne C. Durham,et al.  Closed-form solutions to constrained control allocation problem , 1995 .

[32]  Jonathan Richard Shewchuk,et al.  Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator , 1996, WACG.

[33]  Azriel Rosenfeld,et al.  Learning in Navigation Goal Finding in Graphs , 1996, Int. J. Pattern Recognit. Artif. Intell..

[34]  J. O’Sullivan,et al.  Irregular grain structure in micromagnetic simulation , 1996 .

[35]  Raimund Seidel,et al.  How Good Are Convex Hull Algorithms? , 1997, Comput. Geom..