Congruence Testing of Point Sets in Three and Four Dimensions

Results and Techniques
  • Günter RoteEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9582)


I will survey algorithms for testing whether two point sets are congruent, that is, equal up to an Euclidean isometry. I will introduce the important techniques for congruence testing, namely dimension reduction and pruning, or more generally, condensation. I will illustrate these techniques on the three-dimensional version of the problem, and indicate how they lead for the first time to an algorithm for four dimensions with near-linear running time (joint work with Heuna Kim). On the way, we will encounter some beautiful and symmetric mathematical structures, like the regular polytopes, and Hopf-fibrations of the three-dimensional sphere in four dimensions.


Dimension Reduction Great Circle Coxeter Group Regular Polytopes Orbit Cycle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Akutsu, T.: On determining the congruence of point sets in \(d\) dimensions. Comput. Geom.: Theory Appl. 4(9), 247–256 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Alt, H., Mehlhorn, K., Wagener, H., Welzl, E.: Congruence, similarity, and symmetries of geometric objects. Discrete Comput. Geom. 3(1), 237–256 (1988). MathSciNetCrossRefzbMATHGoogle Scholar
  3. Atkinson, M.D.: An optimal algorithm for geometrical congruence. J. Algorithms 8(2), 159–172 (1987). MathSciNetCrossRefzbMATHGoogle Scholar
  4. Brass, P., Knauer, C.: Testing the congruence of \(d\)-dimensional point sets. Int. J. Comput. Geom. Appl. 12(1–2), 115–124 (2002). MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bentley, J.L., Shamos, M.I.: Divide-and-conquer in multidimensional space. In: Proceedings of the Eighth Annual ACM Symposium on Theory of Computing, STOC 1976, pp. 220–230. ACM, New York (1976).
  6. Conway, J.H., Hardin, R.H., Sloane, N.J.A.: Packing lines, planes, etc.: packings in grassmannian spaces. Exp. Math. 5, 139–159 (1996). MathSciNetCrossRefzbMATHGoogle Scholar
  7. Conway, J.H., Smith, D.A.: On Quaternions and Octonions. A K Peters, Natick (2003)zbMATHGoogle Scholar
  8. Coxeter, H.S.M.: Regular Polytopes, 3rd edn. Dover Publications, New York (1973)zbMATHGoogle Scholar
  9. Hopf, H.: Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche. Math. Ann. 104, 637–665 (1931). MathSciNetCrossRefzbMATHGoogle Scholar
  10. Iwanowski, S.: Testing approximate symmetry in the plane is NP-hard. Theor. Comput. Sci. 80(2), 227–262 (1991). MathSciNetCrossRefzbMATHGoogle Scholar
  11. Kim, H., Rote, G.: Congruence testing of point sets in 4-space. In: Proceedings of the 32st International Symposium on Computational Geometry (SoCG 2016), LIPIcs (2016, to appear)Google Scholar
  12. Manacher, G.: An application of pattern matching to a problem in geometrical complexity. Inf. Process. Lett. 5(1), 6–7 (1976). MathSciNetCrossRefzbMATHGoogle Scholar
  13. Sugihara, K.: An \(n \log n\) algorithm for determining the congruity of polyhedra. J. Comput. Syst. Sci. 29(1), 36–47 (1984). MathSciNetCrossRefzbMATHGoogle Scholar
  14. Threlfall, W., Seifert, H.: Topologische Untersuchung der Diskontinuitätsbereiche endlicher Bewegungsgruppen des dreidimensionalen sphärischen Raumes. Math. Ann. 104(1), 1–70 (1931). MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Institut für InformatikFreie Universität BerlinBerlinGermany

Personalised recommendations