Evolutes and Involutes

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

///////

Related KMR-pages:

Optical properties of Conics

///////

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

///////

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

///////

Ellipse-Evolute:

The interactive simulation that created this movie.

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

///////

EllipseEvolute – Involutes:

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

///////

EllipseParallel Curves:

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

///////

Hyperbola-Evolute:

The interactive simulation that created this movie.

///////

HyperbolaEvolute – Involutes:

The interactive simulation that created this movie.

///////

HyperbolaParallel Curves:

The interactive simulation that created this movie.

///////

Circle-Involutes:

The interactive simulation that created this movie.

///////

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

///////

Explaining the Radian way to measure Angles
(Ambjörn Naeve on YouTube):

///////

CircleInvolute-Evolute:

The interactive simulation that created this movie.

///////

The Catenary as the Evolute of the Tractrix:

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

///////

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.

/////// Quoting Wikipedia:

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.

/////// End of quote from Wikipedia.

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

The interactive simulation that created this movie.

The tractrix as the dog curve of Leibniz at Wikipedia

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

The dog curve of Leibniz (with a card deck)

///////

Cycloid-Evolute:

The interactive simulation that created this movie.

///////

Cycloid-Evolute – Involutes:

The interactive simulation that created this movie.

///////

Epicycloid-Evolute:

The interactive simulation that created this movie.

///////

Hypocycloid-Evolute:

The interactive simulation that created this movie.

///////

Epitrochoid-Evolute:

The interactive simulation that created this movie.

///////

Hypotrochoid-Evolute:

The interactive simulation that created this movie.

///////

Deltroid-Evolute:

The interactive simulation that created this movie.

///////

Limaçon-Evolute:

The interactive simulation that created this movie.

///////

Nephroid-Evolute:

The interactive simulation that created this movie.

///////

Quadrifolium-Evolute:

The interactive simulation that created this movie.

///////

Leave a Reply