Discretizations of the hyperbolic cosine
- 100 Downloads
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.
KeywordsDiscrete differential geometry Discrete catenoid Discrete hyperbolic cosine Discrete tractrix
Mathematics Subject Classification51M04 51M15 52C99 68U05
Unable to display preview. Download preview PDF.
- Bobenko, A.I., Suris, Yu.B.: Discrete differential geometry: integrable structure. In: Graduate Studies in Math., No. 98. American Math. Soc. (2008)Google Scholar
- 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
- Müller, C.: Hexagonal meshes as discrete minimal surfaces. PhD thesis, Technische Universität Graz (2010)Google Scholar
- 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
- Springborn, B., Schröder, P., Pinkall, U.: Conformal equivalence of triangle meshes. ACM Trans. Graphics 27(3), #77, 1–11 (2008)Google Scholar
- Suris, Yu.B.: The Problem of Integrable Discretization: Hamiltonian Approach. Progress in Mathematics, vol. 219. Birkhäuser, Boston (2003)Google Scholar
- Wunderlich, W.: Ebene Kinematik. Bliograph. Inst., Hochschultaschenbücher-Verl (1970)Google Scholar