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Exact Geometric and Algebraic Computations in CGAL

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Mathematical Software – ICMS 2010 (ICMS 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6327))

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Abstract

We summarize recent progress and on-going developments for exact geometric and algebraic computations within the Computational Geometry Algorithms Library (Cgal). We detail the existing machinery used in efficient, exact and robust implementations of various geometric entities.

Partially supported by the FP7-REGPOT-2009-1 project “Archimedes Center for Modeling, Analysis and Computation”.

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References

  1. Berberich, E., Eigenwillig, A., Hemmer, M., Hert, S., Kettner, L., Mehlhorn, K., Reichel, J., Schmitt, S., Schömer, E., Wolpert, N.: EXACUS: Efficient and Exact Algorithms for Curves and Surfaces. In: Proc. 13th European Symposium on Algorithms (ESA). pp. 155–166 (2005)

    Google Scholar 

  2. Berberich, E., Hemmer, M., Kerber, M.: A generic algebraic kernel for non-linear geometric applications. Research Report 7274, INRIA (2010)

    Google Scholar 

  3. Boissonnat, J.D., Devillers, O., Pion, S., Teillaud, M., Yvinec, M.: Triangulations in CGAL. Comput. Geom.-Theor. Appl. 22, 5–19 (2002)

    MATH  MathSciNet  Google Scholar 

  4. Boissonnat, J.D., Teillaud, M. (eds.): Effective Computational Geometry for Curves and Surfaces. Mathematics and Visualization. Springer, Heidelberg (2006)

    Google Scholar 

  5. Brönnimann, H., Burnikel, C., Pion, S.: Interval arithmetic yields efficient dynamic filters for computational geometry. Discrete Appl. Math. 109, 25–47 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  6. de Castro, P.M.M., Cazals, F., Loriot, S., Teillaud, M.: Design of the CGAL 3D spherical kernel and application to arrangements of circles on a sphere. Comput. Geom.-Theor. Appl. 42(6-7), 536–550 (2009)

    MATH  Google Scholar 

  7. de Castro, P.M.M., Pion, S., Teillaud, M.: Exact and efficient computations on circles in CGAL. In: Proc. 23rd European Workshop on Computational Geometry, pp. 219–222. Technische Universität Graz, Austria (2007)

    Google Scholar 

  8. Cgal, Computational Geometry Algorithms Library, http://www.cgal.org

  9. Core Number Library, http://cs.nyu.edu/exact/core_pages

  10. Devillers, O., Teillaud, M.: Perturbations and vertex removal in a 3D Delaunay triangulation. In: Proc. 14th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 313–319 (2003)

    Google Scholar 

  11. Emiris, I.Z., Karavelas, M.I.: The predicates of the Apollonius diagram: algorithmic analysis and implementation. Comput. Geom.-Theor. Appl. 33(1-2), 18–57 (2006)

    MATH  MathSciNet  Google Scholar 

  12. GMP, GNU Multiple Precision Arithmetic Library, http://www.swox.com/gmp/

  13. Hanniel, I., Wein, R.: An exact, complete and efficient computation of arrangements of Bézier curves. In: Proc. ACM Symposium on Solid and Physical Modeling, pp. 253–263 (2007)

    Google Scholar 

  14. Hemmer, M.: Algebraic foundations. In: CGAL User and Reference Manual. CGAL Editorial Board, 3.6 edn. (2010), http://www.cgal.org/Manual/3.6/doc_html/cgal_manual/packages.html#Pkg:AlgebraicFoundations

  15. Karavelas, M.I.: A robust and efficient implementation for the segment Voronoi diagram. In: Proc. International Symposium on Voronoi Diagrams in Science and Engineering (VD 2004), Hongo, Tokyo, Japan, pp. 51–62 (2004)

    Google Scholar 

  16. Kettner, L., Mehlhorn, K., Pion, S., Schirra, S., Yap, C.K.: Classroom examples of robustness problems in geometric computations. Comput. Geom.-Theor. Appl. 40(1), 61–78 (2008)

    MATH  MathSciNet  Google Scholar 

  17. Lazard, S., Peñaranda, L., Tsigaridas, E.P.: Univariate algebraic kernel and application to arrangement. In: Vahrenhold, J. (ed.) SEA 2009. LNCS, vol. 5526, pp. 209–220. Springer, Heidelberg (2009)

    Google Scholar 

  18. LEDA: Library of efficient data-structures and algorithms, http://www.mpi-sb.mpg.de/LEDA/leda.html

  19. Li, C., Pion, S., Yap, C.K.: Recent progress in exact geometric computation. J. Log. Algebr. Program. 64(1), 85–111 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  20. Melquiond, G., Pion, S.: Formally certified floating-point filters for homogeneous geometric predicates. Informatique Théorique et Applications 41(1), 57–69 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  21. The RS library, http://www-salsa.lip6.fr/~rouillie/Software.html

  22. Russel, D., Karavelas, M.I., Guibas, L.J.: A package for exact kinetic data structures and sweepline algorithms. Comp. Geom.-Theor. Appl. 38(1-2), 111–127 (2007)

    MATH  MathSciNet  Google Scholar 

  23. The CGAL Project: CGAL User and Reference Manual. CGAL Editorial Board, 3.6 edn. (2010), http://www.cgal.org/Manual/3.6/doc_html/cgal_manual/packages.html

  24. Wein, R.: High-level filtering for arrangements of conic arcs. In: Möhring, R.H., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 884–895. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  25. Wein, R., Fogel, E., Zukerman, B., Halperin, D.: Advanced programming techniques applied to CGAL’s arrangement package. In: Proc. Library-Centric Software Design Workshop (LCSD 2005), pp. 24–33 (2005)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Dept. Applied Mathematics, University of Crete, GR-714 09, Heraklion, Greece

    Menelaos I. Karavelas

  2. Institute of Applied and Computational Mathematics, Foundation for Research and Technology - Hellas, P.O. Box 1385, GR-711 10, Heraklion, Greece

    Menelaos I. Karavelas

Authors
  1. Menelaos I. Karavelas
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Editor information

Editors and Affiliations

  1. Institute for Operations Research and Institute of Theoretical Computer Science, ETH Zurich, 8092, Zurich, Switzerland

    Komei Fukuda

  2. LIX, CNRS, École polytechnique, 91128, Palaiseau cedex, France

    Joris van der Hoeven

  3. Fachbereich Mathematik, TU Darmstadt, 64289, Darmstadt, Germany

    Michael Joswig

  4. Department of Mathematics, Kobe University, 657-8501, Rokko, Kobe, Japan

    Nobuki Takayama

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Karavelas, M.I. (2010). Exact Geometric and Algebraic Computations in CGAL. In: Fukuda, K., Hoeven, J.v.d., Joswig, M., Takayama, N. (eds) Mathematical Software – ICMS 2010. ICMS 2010. Lecture Notes in Computer Science, vol 6327. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15582-6_20

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  • DOI: https://doi.org/10.1007/978-3-642-15582-6_20

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-15581-9

  • Online ISBN: 978-3-642-15582-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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