Can We Compute the Similarity between Surfaces?

  • Helmut AltEmail author
  • Maike Buchin


A suitable measure for the similarity of shapes represented by parameterized curves or surfaces is the Fréchet distance. Whereas efficient algorithms are known for computing the Fréchet distance of polygonal curves, the same problem for triangulated surfaces is NP-hard. Furthermore, it remained open whether it is computable at all.

Using a discrete approximation, we show that it is upper semi-computable, i.e., there is a non-halting Turing machine which produces a decreasing sequence of rationals converging to the Fréchet distance. It follows that the decision problem, whether the Fréchet distance of two given surfaces lies below a specified value, is recursively enumerable.

Furthermore, we show that a relaxed version of the Fréchet distance, the weak Fréchet distance can be computed in polynomial time. For this, we give a computable characterization of the weak Fréchet distance in a geometric data structure called the Free Space Diagram.


Shape matching Fréchet distance Surfaces 


  1. 1.
    Alt, H., Behrends, B., Blömer, J.: Approximate matching of polygonal shapes. Ann. Math. Artif. Intell. 13, 251–266 (1995) zbMATHCrossRefGoogle Scholar
  2. 2.
    Alt, H., Braß, P., Godau, M., Knauer, C., Wenk, C.: Computing the Hausdorff distance of geometric patterns and shapes. In: Aronov, B., Basu, S., Pach, J., Sharir, M. (eds.) Discrete and Computational Geometry. The Goodman–Pollack Festschrift. Algorithms and Combinatorics, vol. 25, pp. 65–76. Springer, Berlin (2003) Google Scholar
  3. 3.
    Alt, H., Efrat, A., Rote, G., Wenk, C.: Matching planar maps. J. Algorithms 262–283 (2003) Google Scholar
  4. 4.
    Alt, H., Godau, M.: Computing the Fréchet distance between two polygonal curves. Int. J. Comput. Geom. Appl. 5, 75–91 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Buchin, K., Buchin, M., Wenk, C.: Computing the Fréchet distance between simple polygons in polynomial time. In: Proc. 22nd Annu. ACM Symp. Comput. Geom., pp. 80–87. ACM Press, New York (2006) Google Scholar
  6. 6.
    Cole, R.: Slowing down sorting networks to obtain faster sorting algorithms. J. ACM 34(1), 200–208 (1987) CrossRefMathSciNetGoogle Scholar
  7. 7.
    Emiris, I.Z., Tsigaridas, E.P.: Comparing real algebraic numbers of small degree. In: Proc. 12th Annu. European Symp. Algorithms. Lecture Notes Comput. Sci., vol. 3221, pp. 652–663. Springer, Berlin (2004) Google Scholar
  8. 8.
    Fréchet, M.: Sur quelques points du calcul fonctionnel. Rend. Circ. Mat. Palermo 22, 1–74 (1906) CrossRefGoogle Scholar
  9. 9.
    Fréchet, M.: Sur la distance de deux surfaces. Ann. Soc. Pol. Math. 3, 4–19 (1924) Google Scholar
  10. 10.
    Godau, M.: On the complexity of measuring the similarity between geometric objects in higher dimensions. PhD thesis, Freie Universität Berlin, Germany (1998) Google Scholar
  11. 11.
    Megiddo, N.: Applying parallel computation algorithms in the design of serial algorithms. J. ACM 30(4), 852–865 (1983) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Moise, E.E.: Geometric Topology in Dimensions 2 and 3. Graduate Texts in Mathematics, vol. 47. Springer, Berlin (1977) zbMATHGoogle Scholar
  13. 13.
    Mourrain, B., Pion, S., Schmitt, S., Técourt, J.-P., Tsigaridas, E., Wolpert, N.: Algebraic issues in computational geometry. In: Boissonnat, J.-D., Teillaud, M. (eds.) Effective Computational Geometry for Curves and Surfaces. Mathematics and Visualization. Springer, Berlin (2006) Google Scholar
  14. 14.
    van Oostrum, R., Veltkamp, R.C.: Parametric search made practical. Comput. Geom. Theory Appl. 28(2–3), 75–88 (2004) zbMATHGoogle Scholar
  15. 15.
    Weihrauch, K.: Computable Analysis. Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin (2000) zbMATHGoogle Scholar
  16. 16.
    Weihrauch, K., Zheng, X.: Computability on continuous, lower semi-continuous, and upper semi-continuous real functions. Theor. Comput. Sci. 234, 109–133 (2000) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Institut für InformatikFreie Universität BerlinBerlinGermany
  2. 2.Department of Information and Computing SciencesUniversity of UtrechtUtrechtThe Netherlands

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