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In mathematics, an involute (also known as an evolvent) is a particular type of curve that is dependent on another shape or curve. An involute of a curve is the locus of a point on a piece of taut string as the string is either unwrapped from or wrapped around the curve. [1] The evolute of an involute is the original curve.
The evolute of a curve (blue parabola) is the locus of all its centers of curvature (red). The evolute of a curve (in this case, an ellipse) is the envelope of its normals. In the differential geometry of curves, the evolute of a curve is the locus of all its centers of curvature. That is to say that when the center of curvature of each point ...
The involute gear profile, sometimes credited to Leonhard Euler, [1] was a fundamental advance in machine design, since unlike with other gear systems, the tooth profile of an involute gear depends only on the number of teeth on the gear, pressure angle, and pitch. That is, a gear's profile does not depend on the gear it mates with.
Any involution is a bijection.. The identity map is a trivial example of an involution. Examples of nontrivial involutions include negation (x ↦ −x), reciprocation (x ↦ 1/x), and complex conjugation (z ↦ z) in arithmetic; reflection, half-turn rotation, and circle inversion in geometry; complementation in set theory; and reciprocal ciphers such as the ROT13 transformation and the ...
The cycloid, epicycloids, and hypocycloids have the property that each is similar to its evolute. If q is the product of that curvature with the circle's radius, signed positive for epi- and negative for hypo-, then the similitude ratio of curve to evolute is 1 + 2q. The classic Spirograph toy traces out hypotrochoid and epitrochoid curves.
Definitions of evolute, involute, and their relationship. 5-6, 8 Evolute of cycloid and parabola. 7, 9a Rectification of cycloid, semicubical parabola, and history of the problem. 9b-e Circle areas equal to surfaces of conoids; rectification of the parabola equal to quadrature of hyperbola; approximation by logarithms. 10-11
The evolute of a hypocycloid is an enlarged version of the hypocycloid itself, while the involute of a hypocycloid is a reduced copy of itself. [10] The pedal of a hypocycloid with pole at the center of the hypocycloid is a rose curve. The isoptic of a hypocycloid is a hypocycloid.
In the article on Evolute, it is claimed that the involute of an evolute of a curve is the original curve again, and that the evolute of an ellipse is an astroid. These together imply that the involute of an astroid is an ellipse. However, this article claims that the involute of a hypocycloid is a reduced version of the original hypocycloid.