Formally certified floating-point filters for homogeneous geometric predicates
Melquiond, Guillaume ; Pion, Sylvain
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 41 (2007), p. 57-69 / Harvested from Numdam

Floating-point arithmetic provides a fast but inexact way of computing geometric predicates. In order for these predicates to be exact, it is important to rule out all the numerical situations where floating-point computations could lead to wrong results. Taking into account all the potential problems is a tedious work to do by hand. We study in this paper a floating-point implementation of a filter for the orientation-2 predicate, and how a formal and partially automatized verification of this algorithm avoided many pitfalls. The presented method is not limited to this particular predicate, it can easily be used to produce correct semi-static floating-point filters for other geometric predicates.

Publié le : 2007-01-01
DOI : https://doi.org/10.1051/ita:2007005
Classification:  65D18,  65G50,  68Q60
@article{ITA_2007__41_1_57_0,
     author = {Melquiond, Guillaume and Pion, Sylvain},
     title = {Formally certified floating-point filters for homogeneous geometric predicates},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     volume = {41},
     year = {2007},
     pages = {57-69},
     doi = {10.1051/ita:2007005},
     mrnumber = {2330043},
     zbl = {1133.65010},
     language = {en},
     url = {http://dml.mathdoc.fr/item/ITA_2007__41_1_57_0}
}
Melquiond, Guillaume; Pion, Sylvain. Formally certified floating-point filters for homogeneous geometric predicates. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 41 (2007) pp. 57-69. doi : 10.1051/ita:2007005. http://gdmltest.u-ga.fr/item/ITA_2007__41_1_57_0/

[1] H. Brönnimann, C. Burnikel and S. Pion, Interval arithmetic yields efficient dynamic filters for computational geometry. Discrete Appl. Math. 109 (2001) 25-47. | Zbl 0967.68157

[2] C. Burnikel, S. Funke and M. Seel, Exact geometric computation using cascading. Internat. J. Comput. Geom. Appl. 11 (2001) 245-266. | Zbl 1074.65509

[3] The CGAL Manual, 2004. Release 3.1.

[4] M. Daumas and G. Melquiond, Generating formally certified bounds on values and round-off errors, in 6th Conference on Real Numbers and Computers. Dagstuhl, Germany (2004).

[5] O. Devillers and S. Pion, Efficient exact geometric predicates for Delaunay triangulations, in Proc. 5th Workshop Algorithm Eng. Exper. (2003) 37-44.

[6] S. Fortune and C.J. Van Wyk, Static analysis yields efficient exact integer arithmetic for computational geometry. ACM Trans. Graph. 15 (1996) 223-248.

[7] S. Fortune and C.V. Wyk, LN User Manual. AT&T Bell Laboratories (1993).

[8] L. Kettner, K. Mehlhorn, S. Pion, S. Schirra and C. Yap, Classroom examples of robustness problems in geometric computations, in Proc. 12th European Symposium on Algorithms, Lect. Notes Comput. Sci. 3221 (2004) 702-713. | Zbl 1111.68725

[9] A. Nanevski, G. Blelloch and R. Harper, Automatic generation of staged geometric predicates, in International Conference on Functional Programming, Florence, Italy (2001). Also Carnegie Mellon CS Tech Report CMU-CS-01-141.

[10] A. Neumaier, Interval methods for systems of equations. Cambridge University Press (1990). | MR 1100928 | Zbl 0715.65030

[11] S. Pion, De la géométrie algorithmique au calcul géométrique. Thèse de doctorat en sciences, Université de Nice-Sophia Antipolis, France (1999). TU-0619.

[12] J.R. Shewchuk, Adaptive precision floating-point arithmetic and fast robust geometric predicates. Discrete Comput. Geom. 18 (1997) 305-363. | Zbl 0892.68098

[13] D. Stevenson et al., An American national standard: IEEE standard for binary floating point arithmetic. ACM SIGPLAN Notices 22 (1987) 9-25.

[14] C.K. Yap, T. Dubé, The exact computation paradigm, in Computing in Euclidean Geometry, edited by D.-Z. Du and F.K. Hwang, World Scientific, Singapore, 2nd edition, Lect. Notes Ser. Comput. 4 (1995) 452-492.