Computational Geometry from the Viewpoint of Affine Differential Geometry

  • Hiroshi Matsuzoe
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5416)

Abstract

In this paper, we consider Voronoi diagrams from the view point of affine differential geometry. A main object of affine differential geometry is to study hypersurfaces in an affine space that are invariant under the action of the group of affine transformations. Since incidence relations (configurations of vertexes, edges, etc.) in computational geometry are invariant under affine transformations, we may say that affine differential geometry gives a new sight in computational geometry.

The Euclidean distance function can be generalized by a divergence function in affine differential geometry. For such divergence functions, we show that Voronoi diagrams on statistical manifolds are invariant under ( − 1)-conformal transformations. We then give some typical figures of Voronoi diagrams on a manifold. These figures may give good intuition for Voronoi diagrams on a manifold because the figures or constructing algorithms on a manifold strongly depend on the realization or on the choice of local coordinate systems. We also consider the upper envelope type theorems on statistical manifolds, and give a constructing algorithm of Voronoi diagrams on ( − 1)-conformally flat statistical manifolds.

Keywords

information geometry affine differential geometry dually flat space statistical manifold divergence contrast function Voronoi diagram geometric transformation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abe, N.: Affine immersions and conjugate connections. Tensor 55, 276–280 (1994)MathSciNetMATHGoogle Scholar
  2. 2.
    Amari, S., Nagaoka, H.: Method of information geometry. Amer. Math. Soc., Providence. Oxford University Press, Oxford (2000)MATHGoogle Scholar
  3. 3.
    Edelsbrunner, H.: Algorithms in combinatorial geometry. Springer, Berlin (1987)CrossRefMATHGoogle Scholar
  4. 4.
    Eguchi, S.: Geometry of minimum contrast. Hiroshima Math. J. 22, 631–647 (1992)MathSciNetMATHGoogle Scholar
  5. 5.
    Henmi, M., Kobayashi, R.: Hooke’s law in statistical manifolds and divergences. Nagoya Math. J. 159, 1–24 (2000)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Kurose, T.: On the divergences of 1-conformally flat statistical manifolds. Tôhoku Math. J. 46, 427–433 (1994)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Kurose, T.: Conformal-projective geometry of statistical manifolds. Interdiscip. Inform. Sci. 8, 89–100 (2002)MathSciNetMATHGoogle Scholar
  8. 8.
    Kurose, T.: Manifold structures and geometric structures on maximal exponential models. Lecture note for Osaka City University Advanced Mathematical Institute Mini-School on Introduction to Information Geometry and Its Applications II (in Japanese) (2007)Google Scholar
  9. 9.
    Lauritzen, S.L.: Statistical manifolds. In: Differential Geometry in Statistical Inferences. IMS Lecture Notes Monograph Series, vol. 10, pp. 96–163. Institute of Mathematical Statistics, Hayward California (1987)Google Scholar
  10. 10.
    Matsuzoe, H.: On realization of conformally-projectively flat statistical manifolds and the divergences. Hokkaido Math. J. 27, 409–421 (1998)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Matsuzoe, H.: Geometry of contrast functions and conformal geometry. Hiroshima Math. J. 29, 175–191 (1999)MathSciNetMATHGoogle Scholar
  12. 12.
    Matsuzoe, H.: Contrast functions on statistical manifolds with Norden metric. JP J. Geom. Topol. 2, 97–116 (2002)MathSciNetMATHGoogle Scholar
  13. 13.
    Matsuzoe, H.: Voronoi diagrams on ( − 1)-conformally flat statistical manifolds. Far East J. Math. Sci. 4, 235–249 (2002)MathSciNetMATHGoogle Scholar
  14. 14.
    Matsuzoe, H.: Geometry of statistical manifolds and its generalization. In: Proceedings of the 8th International Workshop on Complex Structures and Vector Fields, pp. 244–251. World Scientific, Singapore (2007)Google Scholar
  15. 15.
    Matumoto, T.: Any statistical manifold has a contrast function – On the C 3-function taking the minimum at the diagonal of the product manifold. Hiroshima Math. J. 23, 327–332 (1993)MathSciNetMATHGoogle Scholar
  16. 16.
    Nielsen, F., Boissonnat, J.D., Nock, R.: On Bregman Voronoi diagrams. In: Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms, pp. 746–755 (2007)Google Scholar
  17. 17.
    Nomizu, K., Sasaki, T.: Affine differential geometry – Geometry of Affine Immersions. Cambridge University Press, Cambridge (1994)MATHGoogle Scholar
  18. 18.
    Nomizu, K., Sasaki, T.: Centroaffine immersions of codimension two and projective hypersurface theory. Nagoya Math. J. 132, 63–90 (1993)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Onishi, K., Imai, H.: Voronoi diagrams for an exponential family of probability distributions in information geometry. In: Japan-Korea joint workshop on algorithms and computation, pp. 1–8 (1997)Google Scholar
  20. 20.
    Onishi, K., Itoh, J.: Voronoi diagrams in simply connected complete manifolds. IEICE Transactions E85-A, 944–948 (2002)Google Scholar
  21. 21.
    Onishi, K., Takayama, N.: Construction of Voronoi diagram on the upper half-plane. IEICE Transactions 79-A, 533–539 (1996)Google Scholar
  22. 22.
    Shima, H.: The Geometry of Hessian Structures. World Scientific, Singapore (2007)CrossRefMATHGoogle Scholar
  23. 23.
    Simon, U., Schwenk-Schellschmidt, A., Viesel, H.: Introduction to the affine differential geometry of hypersurfaces. Lecture notes of the Science, University of Tokyo (1991)Google Scholar
  24. 24.
    Zhang, J.: A note on curvature of a-connections of a statistical manifold. Ann. Inst. Stat. Math. 59, 161–170 (2007)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Zhang, J., Matsuzoe, H.: Dualistic Riemannian manifold structure induced from convex function. Advances in Mechanics and Mathematics (to appear)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Hiroshi Matsuzoe
    • 1
  1. 1.Department of Computer Science and Engineering, Graduate School of EngineeringNagoya Institute of TechnologyJapan

Personalised recommendations