Topology-oriented implementation - An approach to robust geometric algorithms

Algorithmica (New York)

Volumn 27, Issue 1, 2000, Pages 5-20

  Sugihara, K ^a   Iri, M ^b   Inagaki, H ^c   Imai, T ^d  

^a UNIVERSITY OF TOKYO   (Japan)

^b CHUO UNIVERSITY   (Japan)

^c TOYOTA NATIONAL COLLEGE OF TECHNOLOGY   (Japan)

^d WAKAYAMA UNIVERSITY   (Japan)

Author keywords

Clipping a convex polyhedron; Line segment voronoi diagram; Robust implementation; Topological consistency

Indexed keywords

EID: 0347248300     PISSN: 01784617     EISSN: None     Source Type: Journal    
DOI: 10.1007/s004530010002     Document Type: Article

References (45)

1. Benouamer, M., Michelucci, D., and Peroche, B., 1993: Error-free boundary evaluation using lazy rational arithmetic - a detailed implementation. Proceedings of the 2nd Symposium on Solid Modeling and Applications, Montreal, May 1993, pp. 115-126.
2. Brönnimann, H., Emiris, I. Z., Pan, V. Y., and Pion, S., 1997: Computing exact geometric predicates using modular arithmetic with single precision. Proceedings of the 13th Annual ACM Symposium on Computational Geometry, Nice, June 1997, pp. 174-182.
3. Brönnimann, H., and Yvinec, M., 1997: Efficient exact evaluation of signs of determinants. Proceedings of the 13th Annual ACM Symposium on Computational Geometry, Nice, June 1997, pp. 166-173.
4. Clarkson, K. L., 1992: Safe and effective determinant evaluation. Proceedings of the 33rd IEEE Symposium on Foundation of Computer Science, pp. 387-395.
5. Dobkin, D., and Silver, D., 1988: Recipes for geometry and numerical analysis - Part 1, An empirical study. Proceedings of the 4th Annual ACM Symposium on Computational Geometry, Urbana-Champaign, June 1988, pp. 93-105.
6. Edelsbrunner, H., and Mücke, E. P., 1988: Simulation of simplicity - A technique to cope with degenerate cases in geometric algorithms. Proceedings of the 4th Annual ACM Symposium on Computational Geometry, Urbana-Champaign, June 1988, pp. 118-133.
7. Fortune, S., 1989: Stable maintenance of point-set triangulations in two dimensions. Proceedings of the 30th IEEE Symposium on Foundations of Computer Science, Research Triangle Park, October 1989, pp. 494-499.
8. Fortune, S., 1995: Numerical stability of algorithms for 2D Delaunay triangulations. International Journal of Computational Geometry and Applications, vol. 5, pp. 193-213.
9. Fortune, S., and Van Wyk, C. J., 1996: Static analysis yields efficient exact integer arithmetic for computational geometry. ACM Transactions on Graphics, vol. 15, pp. 223-248.
10. Greene, D. H., and Yao, F., 1986: Finite-resolution computational geometry. Proceedings of the 27th IEEE Symposium on Foundations of Computer Science, Toronto, October 1986, pp. 143-152.
11. Guibas, L., Salesin, D., and Stolfi, J., 1989: Epsilon geometry - building robust algorithms from imprecise calculations. Proceedings of the 5th Annual ACM Symposium on Computational Geometry, Saarbrücken, June 1989, pp. 208-217.
12. Hiroshima, T., and Sugihara, K., 1996: Recognition of Delaunay graphs. Proceedings of the Korea-Japan Joint Workshop on Algorithms and Computation, Seoul, July 1996, pp. 102-109.
13. Hodgson, C. D., Rivin, I., and Smith, W. D., 1992: A characterization of convex hyperbolic polyhedra and of convex polyhedra inscribed in the sphere. Bulletin of the American Mathematical Society, vol. 27, pp. 246-251.
14. Hoffmann, C. M., 1989: Geometric & Solid Modeling. Morgan Kaufmann, San Mateo.
15. Hoffmann, C. M., Hopcroft, J., and Karasick, M., 1988: Towards implementing robust geometric computations. Proceedings of the 4th Annual ACM Symposium on Computational Geometry, Urbana-Champaign, June 1988, pp. 106-117.
16. Imai, T., 1996: A topology-oriented algorithm for the Voronoi diagram of polygons. Proceedings of the 8th Canadian Conference in Computational Geometry, Ottawa, August 1996, pp. 107-112.
17. Imai, T., 1996: How to get the sign of integers from their residuals. Abstracts of the 9th Franco-Japan Days on Combinatorics and Optimization, p. 7.
18. Imai, T., and Sugihara, K., 1994: A failure-free algorithm for constructing Voronoi diagrams of line segments (in Japanese). Transactions of Information Processing Society of Japan, vol. 35, pp. 1966-1977.
19. Inagaki, H., and Sugihara, K., 1994: Numerically robust algorithm for constructing constrained Delaunay triangulation. Proceedings of the 6th Canadian Conference on Computational Geometry, Saskatoon, August 1994, pp. 171-176.
20. Inagaki, H., Sugihara, K., and Sugie, N., 1992: Numerically robust incremental algorithm for constructing three-dimensional Voronoi diagrams. Proceedings of the 4th Canadian Conference on Computational Geometry, St. John's, August 1992, pp. 334-339.
21. Karasick, M., Lieber, D., and Nackman, L. R., 1989: Efficient Delaunay triangulation using rational arithmetic. IBM Report RC14455, IBM Thomas J. Watson Research Center, Yorktown Heights.
22. Knuth, D. E., 1992: Axioms and Hulls. Lecture Notes in Computer Science, no. 606. Springer-Verlag, Berlin.
23. Liotta, G., Preparata, F. P., and Tamassia, R., 1997: Robust proximity queries-an illustration of degree-driven algorithm design. Proceedings of the 13th Annual ACM Symposium on Computational Geometry, pp. 156-165.
24. Mehlhorn, K., and Näher, S., 1995: A platform for combinatorial and geometric computing. Communications of the ACM, January 1995, pp. 96-102.
25. Milenkovic, V., 1988: Verifiable implementations of geometric algorithms using finite precision arithmetic. Artificial Intelligence, vol. 37, pp. 377-401.
26. Minakawa, T., and Sugihara, K., 1997: Topology-oriented vs. exact arithmetic - experience in implementing three-dimensional convex hull algorithm. In H. W. Leong, H. Imai and S. Jain (eds.): Algorithms and Computation, 8th International Symposium, ISAAC '97, pp. 273-282. Lecture Notes in Computer Science, no. 1350, Springer-Verlag, Berlin.
27. Oishi, Y., and Sugihara, K., 1995: Topology-oriented divide-and-conquer algorithm for Voronoi diagrams. Computer Vision, Graphics, and Image Processing: Graphical Models and Image Processing, vol. 57, pp. 303-314.
28. Ottmann, T., Thiemt, G., and Ullrich, C., 1987: Numerical stability of geometric algorithms. Proceedings of the 3rd Annual ACM Conference on Computational Geometry, Waterloo, June 1987, pp. 119-125.
29. Schorn, P., 1991: Robust algorithms in a program library for geometric computation. Dissertation submitted to the Swiss Federal Institute of Technology, Zürich, for the degree of Doctor of Technical Sciences, Informatik-Dissertationen ETH Zürich, Nr. 32.
30. Segal, M., and Sequin, C. H., 1985: Consistent calculations for solid modeling. Proceedings of the ACM Symposium on Computational Geometry, Baltimore, June 1985, pp. 29-38.
31. Shewchuk, J. R., 1996: Robust adaptive floating-point geometric predicates. Proceedings of the 12th Annual ACM Symposium on Computational Geometry, Philadelphia, May 1996, pp. 141-150.
32. Steinitz, E., 1916: Polyheder und Raumeinteilungen. Encyklopädie der mathematischen Wissenschaften, Band III, Teil 1, 2. Hälfte, IIIAB12, pp. 1-139.
33. Steward, A. J., 1994: Local robustness and its application to polyhedral intersection. International Journal of Computational Geometry and Applications, vol. 4, pp. 87-118.
34. Sugihara, K., 1992: An intersection algorithm based on Delaunay triangulation. IEEE Computer Graphics and Applications, vol. 12, no. 2 (March 1992), pp. 59-67.
35. Sugihara, K., 1993: Approximation of generalized Voronoi diagrams by ordinary Voronoi diagrams. Computer Vision, Graphics, and Image Processing: Graphical Models and Image Processing, vol. 55, pp. 522-531.
36. Sugihara, K., 1994: Robust gift wrapping for the three-dimensional convex hull. Journal of Computer and System Sciences, vol. 49, pp. 391-407.
37. Sugihara, K., 1994: A robust and consistent algorithm for intersecting convex polyhedra. Proceedings of EUROGRAPHICS '94, Oslo, September 1994, pp. C.45-C.54.
38. Sugihara, K., 1997: Experimental study on acceleration of an exact-arithmetic geometric algorithm. International Conference on Shape Modeling and Applications, Aizu-Wakamatsu, March 1997, pp. 160-168.
39. Sugihara, K., and Iri, M., 1988: Geometric algorithms in finite-precision arithmetic. Abstracts of the 13th International Symposium on Mathematical Programming, Tokyo, August-September 1988, WE/3K2, p. 196.
40. Sugihara, K., and Iri, M., 1989: A solid modelling system free from topological inconsistency. Journal of Information Processing, vol. 12, pp. 380-393.
41. Sugihara, K., and Iri, M., 1992: Construction of the Voronoi diagram for "one million" generators in single-precision arithmetic. Proceedings of the IEEE, vol. 80, no. 9, pp. 1471-1484.
42. Sugihara, K., and Iri, M., 1994: A robust topology-oriented incremental algorithm for Voronoi diagrams. International Journal of Computational Geometry and Applications, vol. 4, pp. 179-228.
43. Yap, C., 1988: A geometric consistency theorem for a symbolic perturbation scheme. Proceedings of the 4th Annual ACM Symposium on Computational Geometry, Urbana-Champaign, June 1988, pp. 134-142.
44. Yap, C., 1994: Exact computational geometry and tolerancing metrology. In D. Avis and P. Bose (eds.): Snapshots of Computational Geometry, vol. 3, pp. 34-48. Technical Report SOCS-94.50, McGill University.
45. Zhu, X., Fang, S., Brüderlin, B. D., 1993: Obtaining robust Boolean set operations for manifold solids by avoiding and eliminating redundancy. Proceedings of the 2nd Symposium on Solid Modeling and Applications, Montreal, May 1993, pp. 147-154.