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… 
View on Taylor & Francis
maa.org
6 Citations
Highly Influential Citations
1
Background Citations
5
Methods Citations
1

6 Citations

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.
71st Annual William Lowell Putnam Mathematical Competition

References

SHOWING 1-10 OF 10 REFERENCES
ELEMENTARY CHARACTERIZATION OF CLASSICAL MINIMA
many introductory treatments and is consequently ignored). It is therefore surprising that several optima of classical interest usually treated through the calculus of variations can be characterized…
A Note on the Brachistochrone Problem
(1996). A Note on the Brachistochrone Problem. The College Mathematics Journal: Vol. 27, No. 3, pp. 206-208.
The Brachistochrone Problem
The brachistochrone problem asks for the curve between two points down which an object will slide in the least amount of time under the force of gravity and neglecting friction. Johann Bernoulli…
An Elementary Solution of the Brachistochrone Problem
An introduction to the calculus of variations
A Branch of Mathematics. (Book Reviews: A History of the Calculus of Variations from the 17th through the 19th Century)
The calculus of variations is a subject whose beginning can be precisely dated. It might be said to begin at the moment that Euler coined the name calculus of variations but this is, of course, not…
Johann Bernoulli's brachistochrone solution using Fermat's principle of least time
Johann Bernoulli's brachistochrone problem is now three hundred years old. Bernoulli's solution to the problem he had proposed used the optical analogy of Fermat's least-time principle. In this…
Introduction To The Calculus of Variations And Its Applications
Part 1 Introduction to the problem: examples definition of the most important concepts the question of the existence of solutions. Part 2 The Euler differential equation: derivation of the Euler…
Calculus of Variations, MAA
  • 1925
Bernoulli’s brachistochrone solution using Fermat’s principle of least time, Eur
  • J. Phys
  • 1999