In the
differential geometry of curves, a
roulette is a kind of
curve, generalizing
cycloids,
epicycloids,
hypocycloids,
trochoids, and
involutes.
Roughly speaking, it is the curve described by a point (called the generator or pole) attached to a given curve as it rolls without slipping along a second given curve that is fixed. More precisely, given a curve attached to a plane which is moving so that the curve rolls without slipping along a given curve attached to a fixed plane occupying the same space, then a point attached to the moving plane describes a curve in the fixed plane called a roulette.
In the illustration, the fixed curve (blue) is a parabola, the rolling curve (green) is an equal parabola, and the generator is the vertex of the rolling parabola which describes the roulette (red). In this case the roulette is the cissoid of Diocles.[1]
In the case where the rolling curve is a line and the generator is a point on the line, the roulette is called an involute of the fixed curve. If the rolling curve is a circle and the fixed curve is a line then the roulette is a trochoid. If, in this case, the point lies on the circle then the roulette is a cycloid.
