An approximation of a catenoid constructed from piecewise truncated conical minimal surfaces

TL;DR

This paper presents a novel approximation method for catenoids using piecewise truncated conical minimal surfaces, leveraging hypergeometric functions to closely mimic the continuous surface.

Contribution

It introduces a new approximation technique for catenoids using

Findings

Successfully approximates catenoids with truncated cones

Utilizes hypergeometric functions for precise modeling

Extends previous work to include even truncated cones

Abstract

We consider an appoximation of a catenoid constructed from "odd" truncated cones that maintains minimality in a certain sense. Thorough this procedure, we obtain a discrete curve approximating a catenary by exploiting the fact that it is the function that generates a catenoid. In this investigation, the theory of the Gauss hypergeomtric functions plays an important role. This work is a sequel to [Y.Machigashira, Piecewise truncated conical minimal surfaces and the Gauss hypergeometric functions, Journal of Math-for-Industry 4(2012), pp. 25-33 ]. The paper covers an appoximation of a catenoid constructed from "even" truncated cones that maintains minimality in the same sense. Keywords: catenary, catenoid, truncated cone, hypergeomtric function, Chebyshev polynomial of the third kind.