A Property Characterizing the Catenary

@article{Parker2010APC,
  title={A Property Characterizing the Catenary},
  author={Edward Parker},
  journal={Mathematics Magazine},
  year={2010},
  volume={83},
  pages={63 - 64}
}
  • Edward Parker
  • Published 1 February 2010
  • Mathematics
  • Mathematics Magazine
Summary We show that the area under a catenary curve is proportional to its length in the following sense: given a catenary curve, we can take any horizontal interval and examine the ratio of the area under the curve to the length of the curve on that interval, and we find that the resulting ratio is independent of the chosen interval. This property extends to the three-dimensional case as well: the volume contained by a horizontal interval of a catenoid surface is proportional to its surface… 
VARIOUS CENTROIDS AND SOME CHARACTERIZATIONS OF CATENARIES
For every interval [ ], , b a we denote by ( ) A A y x , and ( ) L L y x , the geometric centroid of the area under a catenary curve = y ( ) ( ) k c x k − cosh defined on this interval and the
A Characteristic Averaging Property of the Catenary
TLDR
A broad characteristic averaging property of the centenary is established that yields two new centroidal characterizations.
Hanging Around in Non-Uniform Fields
TLDR
A family of curves, the n-catenaries, parameterized by the nonzero reals, that include the classical catenaries, parabolas, cycloids, and semicircles are defined.
Various centroids and some characterizations of catenary rotation hypersurfaces
TLDR
This work considers the rectangular domain of positive C^1 functions defined on the n-dimensional Euclidean space with nonzero numbers with coefficients x=(x_1,\cdots, x_n), where $I(x_i)= [0, x-i]$ if $x-i>0 and $I-i= [x- i,0] if $ x- i>0.
Two Generalizations of a Property of the Catenary
TLDR
Two higher-dimensional generalizations of this invariant ratio of the catenary curve are developed and it is found that each invariant ratios identifies a class of hypersurfaces connected to classical objects from differential geometry.

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