Involute

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.

It is generalized by the roulette family of curves. That is, the involutes of a curve are the roulettes of the curve generated by a straight line.

The notions of the involute and evolute of a curve were introduced by Christiaan Huygens in his work titled Horologium oscillatorium sive de motu pendulorum ad horologia aptato demonstrationes geometricae (1673), where he showed that the involute of a cycloid is still a cycloid, thus providing a method for constructing the cycloidal pendulum, which has the useful property that its period is independent of the amplitude of oscillation.^[2]

^ Rutter, J.W. (2000). Geometry of Curves. CRC Press. pp. 204. ISBN 9781584881667.
^ McCleary, John (2013). Geometry from a Differentiable Viewpoint. Cambridge University Press. pp. 89. ISBN 9780521116077.

[1] Rutter, J.W. (2000). Geometry of Curves. CRC Press. pp. 204. ISBN 9781584881667.

[2] McCleary, John (2013). Geometry from a Differentiable Viewpoint. Cambridge University Press. pp. 89. ISBN 9780521116077.

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