Roulette (curve)

In the differential geometry of curves, a roulette is the general concept behind cycloids, epicycloids, hypocycloids, and involutes. Take two curves. Fix some point, called the generator or pole, in relation to the first curve. Roll the first curve along the second; the generator traces out a curve. Such a curve is called a roulette.

Related Topics:
Differential geometry of curves - Cycloid - Epicycloid - Hypocycloid - Involute - Curve - Point

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Working with curves in the complex plane, let r,f:mathbb R omathbb C be parametrisations such that |r'(t)|=|f'(t)| e0 for all t. The roulette of pinmathbb C as r is rolled on f is then

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:tmapsto f(t)+(p-r(t)){f'(t)over r'(t)}

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Roulettes in higher spaces can certainly be imagined but one needs to align more than just the tangents.

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A Sturm roulette traces the center of a conic section as the section rolls on a line.http://www.mathcurve.com/courbes2d/sturm/sturm.shtml

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A Delaunay roulette traces a focus of a conic section as the section rolls on a line.http://www.mathcurve.com/courbes2d/delaunay/delaunay

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