Communications of the
Korean Mathematical Society CKMS

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 $y=k \cosh ((x-c)/k)$ 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 show that one of $\bar{x}_L=\bar{x}_A$ and $\bar{y}_L=2\bar{y}_A$ characterizes the family of catenaries among nonconstant $C^2$ functions. Furthermore, we show that among nonconstant and nonlinear $C^2$ functions, $\bar{y}_L/\bar{x}_L=2\bar{y}_A/\bar{x}_A$ is also a characteristic property of catenaries.

Keywords: centroid, perimeter centroid, area, arc length, catenary

MSC numbers: 52A10, 53A04