The Catenary

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

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Related KMR-pages:

Evolutes and Involutes
Isometric deformations

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Other relevant sources of information:

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

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

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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:

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Catenoid-Helicoid-1:

The interactive simulation that created this movie.

Interactive simulation of Catenoid-Helicoid-2.

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