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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. for all s; see natural parametrization) then the function is a parametrised involute.

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

Examples:

the involute of a catenary through its vertex is a tractrix

With we have and

substitute to get

one involute of a cycloid is a congruent cycloid.

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InvoluteIn 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 l

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