Involute

In the differential geometry of curves, an involute of a smooth curve is another curve, obtained by attaching a string to the curve and tracing the end of the string as it is wound onto the curve. It is a roulette wherein the rolling curve is a straight line containing the generating point.

Analytically: if function r parametrically defines a curve by arc length (i.e. [itex]|r^\prime(s)|=1[itex] for all s; see natural parametrization) then the function [itex]t\mapsto r(t)-tr^\prime(t)[itex] is a parametrised involute.

The evolute of an involute is the original curve less portions of zero or undefined curvature.

Examples:

With [itex]r(s)=(\sinh^{-1}(s),\cosh(\sinh^{-1}(s)))[itex] we have [itex]r^\prime(s)=(1,s)/\sqrt{1+s^2}[itex] and
[itex]r(t)-tr^\prime(t)=(\sinh^{-1}(t)-t/\sqrt{1+t^2},1/\sqrt{1+t^2})[itex]
substitute [itex]t=\sqrt{1-y^2}/y[itex] to get
[itex]({\rm sech}^{-1}(y)-\sqrt{1-y^2},y)[itex]
• one involute of a cycloid is a congruent cycloid.

The involute of a circle has a property that makes it important to the gear industry: if the teeth of two mating gears have the shape of an involute, their relative rates of rotation are constant while the teeth are engaged. With teeth of other shapes, the relative speeds rise and fall as successive teeth engage, resulting in vibration, noise, and excessive wear. For this reason, nearly all modern gear teeth bear the involute shape. See also involute gear.

• Mathworld (http://mathworld.wolfram.com/Involute.html)

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