This page is a sub-page of our page on Plane Curves.
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Related KMR-pages:
• Optical properties of Conics
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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 interactive simulation that created this movie.
• 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:
The interactive simulation that created this movie.
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HyperbolaEvolute – Involutes:
The interactive simulation that created this movie.
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The interactive simulation that created this movie.
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The interactive simulation that created this movie.
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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
(Ambjörn Naeve on YouTube):
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CircleInvolute-Evolute:
The interactive simulation that created this movie.
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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 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.
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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.
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:
The interactive simulation that created this movie.
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The interactive simulation that created this movie.
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Epicycloid-Evolute:
The interactive simulation that created this movie.
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Hypocycloid-Evolute:
The interactive simulation that created this movie.
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Epitrochoid-Evolute:
The interactive simulation that created this movie.
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Hypotrochoid-Evolute:
The interactive simulation that created this movie.
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Deltroid-Evolute:
The interactive simulation that created this movie.
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Limaçon-Evolute:
The interactive simulation that created this movie.
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Nephroid-Evolute:
The interactive simulation that created this movie.
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Quadrifolium-Evolute:
The interactive simulation that created this movie.
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