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Two generalizations of a property of the catenary. (English)
Am. Math. Mon. 121, No. 2, 109-119 (2014).
This paper develops two higher-dimensional generalizations of a well-known property of the catenary curve [{\it E. Parker}, Math. Mag. 83, No. 1, 63‒64 (2010; Zbl 1227.97059)]: The ratio of the area under the catenary to the arc length of the curve is the same over any interval on the horizontal $x-a$ axis. Let $H$ be an $n$-dimensional hypersurface generated by the revolution of the graph of a nonnegative profile $C^2$-function $f(x)$ about the $x$-axis in ${\mathbb R}^{n+1}$ in the sense that the cross-sections of $H$ orthogonal to the $x$-axis are $(n-1)$-spheres of radii $f(x)$. The first generalization is that the ratio of the $(n+1)$-volume enclosed by $H$ to the $n$-volume of $H$ is independent of the chosen horizontal interval if and only if $f(x)$ is the catenary or a constant function; in the two particular cases $n = 1$ and $n = 2$, the “if” part was earlier established by E. Parker [loc. cit.]. The second generalization is that the ratio of the $(n+1)$-volume enclosed by $H$ to the arc length of the curve $f(x)$ is independent of the chosen horizontal interval if and only if $f(x)$ is either $f_n(x)$ or a constant function, where $f_1(x)$ is the catenary function and $f_n(x)$ is a generalization expressed in terms of the incomplete Beta function. It is observed that the hypersurfaces obtained in the first generalization connect to Archimedean ratios. It is also observed that the profile curves $f_n(x)$ obtained in the second generalization are related to minimal hypersurfaces of revolution.
Reviewer: Serge Lawrencenko (Moskva)
Classification: G75 I55