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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. <math>|r^\prime(s)|=1<math> for all s; see natural parametrization) then the function <math>t\mapsto r(t)-tr^\prime(t)<math> is a parametrised involute.
The evolute of an involute is the original curve less portions of zero or undefined curvature.
Examples:
- With <math>r(s)=(\sinh^{-1}(s),\cosh(\sinh^{-1}(s)))<math> we have <math>r^\prime(s)=(1,s)/\sqrt{1+s^2}<math> and
- <math>r(t)-tr^\prime(t)=(\sinh^{-1}(t)-t/\sqrt{1+t^2},1/\sqrt{1+t^2})<math>
- substitute <math>t=\sqrt{1-y^2}/y<math> to get
- <math>({\rm sech}^{-1}(y)-\sqrt{1-y^2},y)<math>
- one involute of a cycloid is a congruent cycloid.
External links
- Mathworld (http://mathworld.wolfram.com/Involute.html)
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