Roulette (curve)

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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.

Working with curves in the complex plane, let $r,f:\mathbb R\to\mathbb C$ be parametrisations such that $|r'(t)|=|f'(t)|\ne0$ for all t. The roulette of $p\in\mathbb C$ as r is rolled on f is then

$t\mapsto f(t)+(p-r(t)){f'(t)\over r'(t)}.$

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

A Sturm roulette traces the center of a conic section as the section rolls on a line.[1] A Delaunay roulette traces a focus of a conic section as the section rolls on a line.[2]

[edit] Example

If the fixed curve is a catenary and the rolling curve is a line, we have:

$f(t)=t+i\cosh(t), \qquad f'(t)=1+i\sinh(t),\,$

$r(t)=\sinh(t), \qquad r'(t)=\cosh(t),\,$

$f(t)+(p-r(t)){f'(t)\over r'(t)}=t+{p-\sinh(t)+i(1+p\sinh(t))\over\cosh(t)}.$

if p = −i the expression is real and the roulette is a horizontal line. In other words, a square wheel could roll without bouncing in a road that was a matched series of catenary arcs.