The Catenary

This page is a sub-page of our page of Plane Curves.


Related KMR-pages:

Evolutes and Involutes
Isometric deformations


Other relevant sources of information:

Catenary at Wikipedia
Catenary at Wolfram Mathworld
Tractrix at Wolfram MathWorld
Catenoid at Wikipedia


The interactive simulations on this page can be navigated with the Free Viewer
of the Graphing Calculator.


Tractrix and Catenary – Involute and Evolute of each other
The catenary is the evolute of the tractrix, and hence
the tractrix is an involute of the catenary:

The interactive simulation that created this movie.

In the movie, the parametric equation of the blue tractrix (of Huygens) is given by

\, x(t) = a \log(\dfrac{1}{\cos 2 \pi t} + \tan 2 \pi t) - a \sin 2 \pi t \,

\, y(t) = a \cos 2 \pi t \, .

The red point is the center of curvature the corresponds to the blue point. As it moves along the tractrix, the red point moves along the light-blue catenary

\, y(x) = a \cosh \dfrac{x}{a} \, ,

which is therefore the evolute of the tractrix. Therefore, the tractrix is the involute of the catenary that corresponds to its vertex point.

/////// Quoting Wikipedia:



The interactive simulation that created this movie.

Interactive simulation of Catenoid-Helicoid-2.


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