CENTROIDS AND SOME CHARACTERIZATIONS OF CATENARIES

Abstract

For every interval [a, b], we denote by ((x) over bar (A), (y) over bar (A)) and ((x) over bar (L), (y) over bar (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 (x) over bar (L) = (x) over bar (A) and (y) over bar (L) = 2 (y) over bar (A). In this paper, we show that one of ($) over bar (L) = (x) over bar (A) and (y) over bar (L) = 2 (y) over bar (A) characterizes the family of catenaries among nonconstant C-2 functions. Furthermore, we show that among nonconstant and nonlinear C-2 functions, (y) over bar (L)/(x) over bar (L) = 2 (y) over bar (A)/(x) over bar (A) is also a characteristic property of catenaries.