Commun. Korean Math. Soc. 2018; 33(1): 237-245
Online first article June 14, 2017 Printed January 31, 2018
https://doi.org/10.4134/CKMS.c170041
Copyright © The Korean Mathematical Society.
Shin-Ok Bang, Dong-Soo Kim, Dae Won Yoon
Chonnam National University, Chonnam National University, Gyeongsang National University
For every interval $[a,b]$, we denote by $(\bar{x}_A, \bar{y}_A)$ and $(\bar{x}_L, \bar{y}_L)$ the geometric centroid of the area under a catenary curve $y=k \cosh ((x-c)/k)$ defined on this interval and the centroid of the curve itself, respectively. Then, it is well-known that $\bar{x}_L=\bar{x}_A$ and $\bar{y}_L=2\bar{y}_A$. In this paper, we fix an end point, say $0$, and we show that one of $\bar{x}_L=\bar{x}_A$ and $\bar{y}_L=2\bar{y}_A$ for every interval with an end point $0$ characterizes the family of catenaries among nonconstant $C^2$ functions.
Keywords: centroid, perimeter centroid, area, arc length, catenary
MSC numbers: 52A10, 53A04
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