Discretizations of the hyperbolic cosine
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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
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