Involute

Attach a string to a point on a curve. Extend the string so that it is tangent to the curve at the point of attachment. Then wind the string up, keeping it always taut. The locus of points traced out by the end of the string is called the involute of the original curve, and the original curve is called the evolute of its involute. This process is illustrated above for a circle.

Although a curve has a unique evolute, it has infinitely many involutes corresponding to different choices of initial point. An involute can also be thought of as any curve orthogonal to all the tangents to a given curve.

The equation of the involute is

(1)

where is the tangent vector

(2)

and is the arc length

(3)

This can be written for a parametrically represented function as

			(4)
			(5)

The following table lists the involutes of some common curves, some of which are illustrated above.

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