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

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Related KMR-pages:

• Evolutes and Involutes
• Hyperbolic geometry

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Other relevant sources of information:

• Tractrix at Wikipedia
• Tractrix at Wolfram Mathworld

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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.

• The Tractrix (at Wolfram MathWorld)

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.

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Practical Applications:

In 1927, P. G. A. H. Voigt patented a horn loudspeaker design based on the assumption that a wave front traveling through the horn is spherical of a constant radius. The idea is to minimize distortion caused by internal reflection of sound within the horn. The resulting shape is the surface of revolution of a tractrix.

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The Tractrix as the “Dog Curve” of Leibniz
(See: The tractrix at Wolfram MathWorld)

The interactive simulation that created this movie.

An approximation (using a deck of cards) of the dog curve of Leibniz:

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