# Evolutes and Involutes

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Related KMR-pages:

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Other related sources of information:

Evolutes at Wikipedia
Involutes at Wikipedia
Involutes at Wikipedia/wiki/Media
Evolutes and Involutes at Math24
Parallel Curves at Wikipedia
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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Ellipse-Evolute:

The Evolute of an Ellipse at Wolfram MathWorld
The Involutes of an Ellipse at Wolfram MathWorld

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EllipseEvolute – Involutes:

The interactive simulation that created this movie.
Drag the purple point at the bottom in order to change the involute.

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The interactive simulation that created this movie.
Drag the purple point at the bottom to change the involute.

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Hyperbola-Evolute:

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HyperbolaEvolute – Involutes:

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A Circleinvolute unwinding a string from the circle starting from the angle 0:

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Explaining the Radian way to measure Angles

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CircleInvolute-Evolute:

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The Catenary as the Evolute of the Tractrix:

The Tractrix as the Involute of the Catenary that corresponds to its vertex point:

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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 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 tractrix as the dog curve of Leibniz at Wikipedia

Using a deck of cards to approximate the dog curve of Leibniz:

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Cycloid-Evolute:

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Cycloid-Evolute – Involutes:

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Epicycloid-Evolute:

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Hypocycloid-Evolute:

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Epitrochoid-Evolute:

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Hypotrochoid-Evolute:

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Deltroid-Evolute:

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Limaçon-Evolute:

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Nephroid-Evolute:

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