Qualitative Symbolic Perturbation

Authors Olivier Devillers, Menelaos Karavelas, Monique Teillaud

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Olivier Devillers
Menelaos Karavelas
Monique Teillaud

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Olivier Devillers, Menelaos Karavelas, and Monique Teillaud. Qualitative Symbolic Perturbation. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 33:1-33:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


In a classical Symbolic Perturbation scheme, degeneracies are handled by substituting some polynomials in epsilon for the inputs of a predicate. Instead of a single perturbation, we propose to use a sequence of (simpler) perturbations. Moreover, we look at their effects geometrically instead of algebraically; this allows us to tackle cases that were not tractable with the classical algebraic approach.
  • Robustness issues
  • Symbolic perturbations
  • Apollonius diagram


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  1. P. Alliez, O. Devillers, and J. Snoeyink. Removing degeneracies by perturbing the problem or the world. Reliable Computing, 6:61-79, 2000. URL: http://hal.inria.fr/inria-00338566/.
  2. H. Brönnimann, O. Devillers, V. Dujmović, H. Everett, M. Glisse, X. Goaoc, S. Lazard, H.-S. Na, and S. Whitesides. Lines and free line segments tangent to arbitrary three-dimensional convex polyhedra. SIAM Journal on Computing, 37:522-551, 2007. URL: http://hal.inria.fr/inria-00103916.
  3. C. Burnikel, K. Mehlhorn, and S. Schirra. On degeneracy in geometric computations. In 5th ACM-SIAM Sympos. Discrete Algorithms, pages 16-23, 1994. URL: http://dl.acm.org/citation.cfm?id=314474.
  4. O. Devillers, M. Glisse, and S. Lazard. Predicates for line transversals to lines and line segments in three-dimensional space. In Proc. 24th Annual Symposium on Computational Geometry, pages 174-181, 2008. URL: http://hal.inria.fr/inria-00336256/.
  5. O. Devillers, M. Karavelas, and M. Teillaud. Qualitative symbolic perturbation: two applications of a new geometry-based perturbation framework. Research Report 8153, INRIA, 2015. version 4. URL: http://hal.inria.fr/hal-00758631/.
  6. O. Devillers and M. Teillaud. Perturbations for Delaunay and weighted Delaunay 3D triangulations. Computational Geometry: Theory and Applications, 44:160-168, 2011. URL: http://dx.doi.org/10.1016/j.comgeo.2010.09.010.
  7. H. Edelsbrunner and E. P. Mücke. Simulation of simplicity: A technique to cope with degenerate cases in geometric algorithms. ACM Trans. Graph., 9(1):66-104, 1990. URL: http://dl.acm.org/citation.cfm?id=77639.
  8. I. Emiris and J. Canny. A general approach to removing degeneracies. SIAM J. Comput., 24:650-664, 1995. URL: http://epubs.siam.org/sicomp/resource/1/smjcat/v24/i3/p650_s1.
  9. I. Emiris and M. Karavelas. The predicates of the Apollonius diagram: algorithmic analysis and implementation. Computational Geometry: Theory and Applications, 33(1-2):18-57, January 2006. URL: http://dx.doi.org/10.1016/j.comgeo.2004.02.006.
  10. G. Irving and F. Green. A deterministic pseudorandom perturbation scheme for arbitrary polynomial predicates. Technical Report 1308.1986v1, arXiv, 2013. URL: http://arxiv.org/abs/1308.1986.
  11. K. Mehlhorn, R. Osbild, and M. Sagraloff. A general approach to the analysis of controlled perturbation algorithms. Comput. Geom. Theory Appl., 44:507-528, 2011. URL: http://dx.doi.org/10.1016/j.comgeo.2011.06.001.
  12. R. Seidel. The nature and meaning of perturbations in geometric computing. Discrete Comput. Geom., 19:1-17, 1998. Google Scholar
  13. R. Seidel. Perturbations in geometric computing, 2013. Workshop on Geometric Computing, Heraklion. URL: http://www.acmac.uoc.gr/GC2013/files/Seidel-slides.pdf.
  14. C. K. Yap. A geometric consistency theorem for a symbolic perturbation scheme. J. Comput. Syst. Sci., 40(1):2-18, 1990. URL: http://www.sciencedirect.com/science/article/pii/002200009090016E.
  15. C. K. Yap. Symbolic treatment of geometric degeneracies. J. Symbolic Comput., 10:349-370, 1990. URL: http://www.sciencedirect.com/science/article/pii/S0747717108800697.
  16. C. K. Yap and T. Dubé. The exact computation paradigm. In Computing in Euclidean Geometry, volume 4 of Lecture Notes Series on Computing, pages 452-492. World Scientific, 1995. URL: http://www.cs.nyu.edu/~exact/doc/paradigm.ps.gz.