Discretizations of the hyperbolic cosine

Original Paper

Abstract

This paper explores different notions of a discrete hyperbolic cosine. The interest in this topic arises from the discretization of the catenoid which is a minimal surface of revolution and whose meridian curve is the hyperbolic cosine. Different but equivalent characterizations of the smooth hyperbolic cosine function lead to different discretizations which are no longer equivalent. However, it turns out that there are still some interrelations. We are led to some explicit and recursive definitions. It is also natural that we study discretizations of the tractrix whose evolute is the hyperbolic cosine, and its relation to discrete surfaces of constant Gaussian curvature. We can show convergence results for the discrete hyperbolic cosine and the discrete tractrix to their smooth counterparts.

Keywords

Discrete differential geometry Discrete catenoid Discrete hyperbolic cosine Discrete tractrix 

Mathematics Subject Classification

51M04 51M15 52C99 68U05 

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References

  1. Bobenko A.I., Pinkall U.: Discrete isothermic surfaces. J. Reine Angew. Math. 475, 187–208 (1996)MathSciNetMATHGoogle Scholar
  2. Bobenko, A.I., Suris, Yu.B.: Discrete differential geometry: integrable structure. In: Graduate Studies in Math., No. 98. American Math. Soc. (2008)Google Scholar
  3. Bobenko A.I., Hoffmann T., Springborn B.: Minimal surfaces from circle patterns: geometry from combinatorics. Ann. Math. 164, 231–264 (2006)MathSciNetCrossRefMATHGoogle Scholar
  4. Bobenko A.I., Pottmann H., Wallner J.: A curvature theory for discrete surfaces based on mesh parallelity. Math. Annalen 348, 1–24 (2010)MathSciNetCrossRefMATHGoogle Scholar
  5. Chorlton F.: Some geometrical properties of the catenary. Math. Gazette 83(496), 121–123 (1999)CrossRefGoogle Scholar
  6. Christoffel E.B.: Ueber einige allgemeine Eigenschaften der Minimumsflächen. J. Reine Angew. Math. 67, 218–228 (1867)CrossRefMATHGoogle Scholar
  7. Hoffmann, T.: On discrete differential geometry and its links to visualization. In: Consortium “Math for Industry” First Forum, COE Lect. Note, vol. 12, pp. 45–51. Kyushu Univ. Fac. Math., Fukuoka (2008)Google Scholar
  8. Kruppa E.: Analytische und konstruktive Differentialgeometrie. Springer, Wien (1957)CrossRefMATHGoogle Scholar
  9. Müller, C.: Hexagonal meshes as discrete minimal surfaces. PhD thesis, Technische Universität Graz (2010)Google Scholar
  10. Müller C.: Conformal hexagonal meshes. Geometriae Dedicata 154, 27–46 (2011)MathSciNetCrossRefMATHGoogle Scholar
  11. Müller C., Wallner J.: Oriented mixed area and discrete minimal surfaces. Discrete Comput. Geom. 43, 303–320 (2010)MathSciNetCrossRefMATHGoogle Scholar
  12. Pinkall U., Polthier K.: Computing discrete minimal surfaces and their conjugates. Exp. Math. 2, 15–36 (1993)MathSciNetCrossRefMATHGoogle Scholar
  13. Pottmann, H., Liu, Y., Wallner, J., Bobenko, A.I., Wang, W.: Geometry of multi-layer freeform structures for architecture. ACM Trans. Graphics 26(3), #65, 1–11 (2007)Google Scholar
  14. Sauer R.: Differenzengeometrie. Springer, Berlin (1970)CrossRefMATHGoogle Scholar
  15. Springborn, B., Schröder, P., Pinkall, U.: Conformal equivalence of triangle meshes. ACM Trans. Graphics 27(3), #77, 1–11 (2008)Google Scholar
  16. Suris, Yu.B.: The Problem of Integrable Discretization: Hamiltonian Approach. Progress in Mathematics, vol. 219. Birkhäuser, Boston (2003)Google Scholar
  17. Wunderlich, W.: Ebene Kinematik. Bliograph. Inst., Hochschultaschenbücher-Verl (1970)Google Scholar

Copyright information

© The Managing Editors 2012

Authors and Affiliations

  1. 1.Institute of Discrete Mathematics and Geometry, Vienna University of TechnologyViennaAustria

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