Advertisement

Geometriae Dedicata

, Volume 182, Issue 1, pp 95–116 | Cite as

A topological classification of convex bodies

  • Gábor Domokos
  • Zsolt Lángi
  • Tímea Szabó
Original Paper

Abstract

The shape of homogeneous, generic, smooth convex bodies as described by the Euclidean distance with nondegenerate critical points, measured from the center of mass represents a rather restricted class \({\mathcal {M}}_C\) of Morse–Smale functions on \({\mathbb {S}}^2\). Here we show that even \({\mathcal {M}}_C\) exhibits the complexity known for general Morse–Smale functions on \({\mathbb {S}}^2\) by exhausting all combinatorial possibilities: every 2-colored quadrangulation of the sphere is isomorphic to a suitably represented Morse–Smale complex associated with a function in \({\mathcal {M}}_C\) (and vice versa). We prove our claim by an inductive algorithm, starting from the path graph \(P_2\) and generating convex bodies corresponding to quadrangulations with increasing number of vertices by performing each combinatorially possible vertex splitting by a convexity-preserving local manipulation of the surface. Since convex bodies carrying Morse–Smale complexes isomorphic to \(P_2\) exist, this algorithm not only proves our claim but also generalizes the known classification scheme in Várkonyi and Domokos (J Nonlinear Sci 16:255–281, 2006). Our expansion algorithm is essentially the dual procedure to the algorithm presented by Edelsbrunner et al. (Discrete Comput Geom 30:87–10, 2003), producing a hierarchy of increasingly coarse Morse–Smale complexes. We point out applications to pebble shapes.

Keywords

Equilibrium Convex surface Morse–Smale complex  Vertex splitting Quadrangulation Pebble shape 

Mathematics Subject Classification (2010)

52A15 53A05 53Z05 

Notes

Acknowledgments

This research was supported by OTKA grant T104601. The authors thank an anonymous referee for suggesting substantial improvements to the paper. The authors are indebted to E. Makai Jun., G. Etesi and Sz. Szabó for their valuable comments on smooth approximations of continuous functions. Z. Lángi also acknowledges the support of the Fields Institute for Research in Mathematical Sciences, University of Toronto, Toronto ON, Canada, and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences.

References

  1. 1.
    Andreev, E.M.: Convex polyhedra in Lobachevsky spaces. Mat. Sb. (N.S.) 81(123), 445–478 (1970)MathSciNetGoogle Scholar
  2. 2.
    Archdeacon, D., Hutchinson, J., Nakamoto, A., Negami, S., Ota, K.: Chromatic numbers of quadrangulations on closed surfaces. J. Graph Theory 37, 100–114 (2001)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Arnold, V.I.: Ordinary Differential Equations, 10th printing. MIT Press, Cambridge (1998)Google Scholar
  4. 4.
    Arnold, V.I. (ed.): Dynamical Systems V: Bifurcation Theory and Catastrophe Theory. Springer, Berlin (1994)Google Scholar
  5. 5.
    Bagatelj, V.: An inductive definition of the class of 3-connected quadrangulations of the plane. Discrete Math. 78, 45–53 (1989)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bauer, U., Lange, C., Wardetzky, M.: Optimal topological simplification of discrete functions on surfaces. Discrete Comput. Geom. 47, 347–377 (2012)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bonnesen, T., Fenchel, W.: Theory of Convex Bodies. BCS Associates, Moscow (1987)Google Scholar
  8. 8.
    Bloore, F.J.: The shape of pebbles. Math. Geol. 9, 113–122 (1977)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bremer, P.T., Edelsbrunner, H., Hamann, B., Pascucci, V.: A multi-resolution data structure for two-dimensional Morse–Smale functions. In: Proceeding VIS ’03, pp. 139–146 (2003)Google Scholar
  10. 10.
    Brinkmann, G., Greenberg, S., Greenhill, C., McKay, B.D., Thomas, R., Wollan, P.: Generation of simple quadrangulations of the sphere. Discrete Math. 305, 33–54 (2005)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Conway, J.H., Guy, R.: Stability of polyhedra. SIAM Rev. 11, 78–82 (1969)CrossRefGoogle Scholar
  12. 12.
    Dawson, R.: Monostatic simplexes. Amer. Math. Month. 92, 541–546 (1985)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Dawson, R., Finbow, W.: What shape is a loaded die? Math. Intell. 22, 32–37 (1999)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Dey, T.K., Li, K., Luo, C., Ranjan, P., Safa, I., Wang, Y.: Persistent heat signature for pose-oblivious matching of incomplete models. Comput. Graph. Forum 29, 1545–1554 (2010)CrossRefGoogle Scholar
  15. 15.
    Diestel, R.: Graph Theory, 3rd edn. Springer, Heidelberg (2005)MATHGoogle Scholar
  16. 16.
    Domokos, G., Lángi, Z., Szabó, T.: On the equilibria of finely discretized curves and surfaces. Monatsh. Math. 168, 321–345 (2012)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Domokos, G., Sipos, A.Á., Szabó, T.: The mechanics of rocking stones: equilibria on separated scales. Math. Geosci. 44, 71–89 (2012)CrossRefGoogle Scholar
  18. 18.
    Domokos, G., Sipos, A.Á., Várkonyi, P.: Continuous and discrete models for abrasion processes. Period. Polytech. Arch. 40, 3–8 (2009)CrossRefGoogle Scholar
  19. 19.
    Dong, S., Bremer, P.-T., Garland, M., Pascucci, V., Hart, J.C.: Spectral surface quadrangulation. ACM Trans. Graph. 25, 1057–1066 (2006)CrossRefGoogle Scholar
  20. 20.
    Edelsbrunner, H., Harer, J., Zomorodian, A.: Hierarchical Morse–Smale complexes for piecewise linear 2-manifolds. Discrete Comput. Geom. 30, 87–107 (2003)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Evans, L.: Partial Differential Equations, Graduate Texts in Mathematics 19. American Mathematical Society, Providence, RI (1998)Google Scholar
  22. 22.
    Ghomi, M.: The problem of optimal smoothing for convex functions. Proc. Amer. Math. Soc. 130, 2255–2259 (2002)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Gross, J.T., Yellen, J.: Graph Theory and Its Applications, 2nd edn. CRC Press, Boca Raton, FL (2006)MATHGoogle Scholar
  24. 24.
    Gyulassy, A., Natarajan, V., Pascucci, V., Hamann, B.: Efficient computation of Morse–Smale complexes for three-dimensional scalar functions. IEEE Trans. Vis. Comput. Graph. 13, 1440–1447 (2007)CrossRefGoogle Scholar
  25. 25.
    Heath, T.I. (ed.): The Works of Archimedes. Cambridge University Press, Cambridge (1897)Google Scholar
  26. 26.
    Heppes, A.: A double-tipping tetrahedron. SIAM Rev. 9, 599–600 (1967)CrossRefGoogle Scholar
  27. 27.
    Hirsch, M.: Differential Topology, Graduate Texts in Mathematics 33. Springer, New York-Heidelberg (1976)Google Scholar
  28. 28.
    Kápolnai, R., Domokos, G.: Inductive generation of convex bodies. In: The 7th Hungarian–Japanese Symposium on Discrete Mathematics and Its Applications, pp. 170–178 (2011)Google Scholar
  29. 29.
    Krapivsky, P.L., Redner, S.: Smoothing a rock by chipping. Phys. Rev. E 9 75(3 Pt 1), 031119 (2007)CrossRefGoogle Scholar
  30. 30.
    Negami, S., Nakamoto, A.: Diagonal transformations of graphs on closed surfaces. Sci. Rep. Yokohama Nat. Univ., Sec. I 40, 71–97 (1993)MathSciNetGoogle Scholar
  31. 31.
    Poston, T., Stewart, J.: Catastrophe Theory and its Applications. Pitman, London (1978)MATHGoogle Scholar
  32. 32.
    Roeder, R.K.W., Hubbard, J.H., Dunbar, W.D.: Andreev’s theorem on hyperbolic polyhedra. Ann. Inst. Fourier (Grenoble) 57(3), 825–882 (2007)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Sipos, A.Á., Domokos, G., Wilson, A., Hovius, N.: A discrete random model describing bedrock erosion. Math. Geosci. 43, 583–591 (2011)CrossRefGoogle Scholar
  34. 34.
    Várkonyi, P.L., Domokos, G.: Static equilibria of rigid bodies: dice, pebbles and the Poincaré–Hopf Theorem. J. Nonlinear Sci. 16, 255–281 (2006)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Zamfirescu, T.: How do convex bodies sit? Mathematica 42, 179–181 (1995)MathSciNetMATHGoogle Scholar
  36. 36.
    Zomorodian, A.: Topology for Computing. Cambridge University Press, Cambridge (2005)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of Mechanics, Materials and StructuresBudapest University of TechnologyBudapestHungary
  2. 2.Department of GeometryBudapest University of TechnologyBudapestHungary

Personalised recommendations