Experimental study on acceleration of an exact-arithmetic geometric algorithm

The paper presents a method for accelerating an exact arithmetic geometric algorithm. The exact arithmetic is one of the most promising approaches for making numerically robust geometric algorithms, because it enables us to always judge the topological structures of objects correctly and thus makes us free from inconsistency. However, exact arithmetic costs much more time than floating point arithmetic. In order to decrease this cost, the paper studies a hybrid method using both exact and floating point arithmetic. For each judgement in the algorithm, floating point arithmetic is first applied, and exact arithmetic is used only when the floating point computation is not reliable. This idea is applied to the construction of three dimensional convex hulls, and experiments show that 80/spl sim/95% of the computational cost can be saved.

[1]  David P. Dobkin,et al.  Recipes for geometry and numerical analysis - Part I: an empirical study , 1988, SCG '88.

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

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

[4]  B. Peroche,et al.  Error-free boundary evaluation using lazy rational arithmetic: a detailed implementation , 1993, Solid Modeling and Applications.

[5]  Carlo H. Séquin,et al.  Consistent calculations for solids modeling , 1985, SCG '85.

[6]  Kokichi Sugihara.  A Robust and Consistent Algorithm for Intersecting Convex Polyhedra , 1994, Comput. Graph. Forum.

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

[8]  Peter Schorn.  Robust algorithms in a program library for geometric computation , 1991, Informatik-Dissertationen ETH Zürich.

[9]  Chee-Keng Yap,et al.  A geometric consistency theorem for a symbolic perturbation scheme , 1988, SCG '88.

[10]  Christopher J. Van Wyk,et al.  Efficient exact arithmetic for computational geometry , 1993, SCG '93.

[11]  Chee Yap.  Symbolic treatment of geometric degeneracies: Proceedings of the International IFIPS Conference on System Modeling and Optimization. Tokyo, 1987 , 1987 .

[12]  Franz Aurenhammer,et al.  Power Diagrams: Properties, Algorithms and Applications , 1987, SIAM J. Comput..

[13]  Kokichi Sugihara.  Approximation of Generalized Voronoi Diagrams by Ordinary Voronoi Diagrams , 1993, CVGIP Graph. Model. Image Process..

[14]  Christoph M. Hoffmann,et al.  The problems of accuracy and robustness in geometric computation , 1989, Computer.

[15]  Kokichi Sugihara,et al.  A solid modelling system free from topological inconsistency , 1990 .

[16]  Thomas Ottmann,et al.  Numerical stability of geometric algorithms , 1987, SCG '87.

[17]  Donald E. Knuth,et al.  Axioms and Hulls , 1992, Lecture Notes in Computer Science.

[18]  M HoffmannChristoph.  The Problems of Accuracy and Robustness in Geometric Computation , 1989 .

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

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

[21]  Lee R. Nackman,et al.  Efficient Delaunay triangulation using rational arithmetic , 1991, TOGS.

[22]  Atsuyuki Okabe,et al.  Spatial Tessellations: Concepts and Applications of Voronoi Diagrams , 1992, Wiley Series in Probability and Mathematical Statistics.

[23]  Chee-Keng Yap,et al.  Towards Exact Geometric Computation , 1997, Comput. Geom..

[24]  Hiroshi Imai,et al.  Voronoi Diagram in the Laguerre Geometry and its Applications , 1985, SIAM J. Comput..

[25]  Christoph M. Hoffmann,et al.  Geometric and Solid Modeling: An Introduction , 1989 .