Robust Construction of the Voronoi Diagram of a Polyhedron

This paper describes a practical algorithm for the construction of the Voronoi diagram of a three dimensional polyhedron using approximate arithmetic. This algorithm is intended to be implemented in oating point arithmetic. The full two-dimensional version and sig-niicant portions of the three-dimensional version have been implemented and tested. The running time 1 of this algorithm is O(npnv log 2 b), where np is the size of the input polyhedron, nv is the size of the output Voronoi diagram, and b is the number of desired bits of precision. This algorithm can be made more practical with the use of a binary partition of space. In the worst case, binary partition does not improve the running time, but it should reduce the running time to O(nvb) on well-behaved inputs. Since b is constant, this eliminates a factor of np. The algorithm can be generalized to higher dimensions and the order k Voronoi diagram.

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

[2]  Joseph S. B. Mitchell,et al.  Separation and approximation of polyhedral surfaces , 1991 .

[3]  M. Iri,et al.  Construction of the Voronoi diagram for 'one million' generators in single-precision arithmetic , 1992, Proc. IEEE.

[4]  Micha Sharir,et al.  Planning a purely translational motion for a convex object in two-dimensional space using generalized Voronoi diagrams , 2016, Discret. Comput. Geom..

[5]  D. T. Lee,et al.  Generalization of Voronoi Diagrams in the Plane , 1981, SIAM J. Comput..

[6]  Michael Ian Shamos,et al.  Closest-point problems , 1975, 16th Annual Symposium on Foundations of Computer Science (sfcs 1975).

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

[8]  Victor J. Milenkovic.  Calculating approximate curve arrangements using rounded arithmetic , 1989, SCG '89.

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

[10]  Victor J. Milenkovic,et al.  Verifiable Implementations of Geometric Algorithms Using Finite Precision Arithmetic , 1989, Artif. Intell..

[11]  Victor J. Milenkovic,et al.  Double precision geometry: a general technique for calculating line and segment intersections using rounded arithmetic , 1989, 30th Annual Symposium on Foundations of Computer Science.

[12]  Victor J. Milenkovic,et al.  Numerical stability of algorithms for line arrangements , 1991, SCG '91.

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

[14]  Der-Tsai Lee.  On k-Nearest Neighbor Voronoi Diagrams in the Plane , 1982, IEEE Transactions on Computers.

[15]  John E. Hopcroft,et al.  Towards implementing robust geometric computations , 1988, SCG '88.

[16]  Steven Fortune,et al.  A sweepline algorithm for Voronoi diagrams , 1986, SCG '86.

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

[18]  Leonidas J. Guibas,et al.  Approximating Polygons and Subdivisions with Minimum Link Paths , 1991, ISA.