Roulette (curve)

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Construction of a roulette, in fact a cissoid of Diocles.

In the differential geometry of curves, a roulette is a kind of curve, generalizing cycloids, epicycloids, hypocycloids, 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.

More precisely, given a curve attached to a plane which is moving so that the curve rolls without slipping along a given fixed curve, 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.

If, instead of a single point being attached to the rolling curve, another given curve is carried along the moving plane, a family of congruent curves is produced. The envelope of this family may also be called an roulette.

A related concept is a glissette, the curve described by a point attached to a given curve as it slides along two (or more) given curves.

Formally speaking, the curves must be differentiable curves in the Euclidean plane. One is kept invariant; the other is subjected to a continuous congruence transformation such that at all times the curves are tangent at a point of contact that moves with the same speed when taken along either curve. The resulting roulette is formed by the locus of the generator subjected to the same set of congruence transformations.

Modelling the original curves as curves in the complex plane, let $r,f:\mathbb R\to\mathbb C$ be differentiable parametrisations such that $r (0) = f (0),$ $r'(0) = f'(0),$ and $|r'(t)|=|f'(t)|\ne0$ for all t. The roulette of $p\in\mathbb C$ as r is rolled on f is then given by the mapping:

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

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

1 Example
2 List of roulettes
3 Notes
4 References
5 Further reading

[edit] Example

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

$f(t)=t+i\cosh(t) \qquad r(t)=\sinh(t)$

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

The parameterization of the line is chosen so that

$|f'(t)| \,$ $=\sqrt{1^2+\sinh^2(t)}$ $=\sqrt{\cosh^2(t)}$ $=|r'(t)| \,$

Applying the formula above we obtain:

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

If p = −i the expression is real (namely t) and the roulette is a horizontal line. An interesting application of this is that a square wheel could roll without bouncing in a road that was a matched series of catenary arcs.

[edit] List of roulettes

Fixed Curve	Rolling Curve	Generating Point	Roulette
Any curve	Line	Point on the line	Involute of the the curve
Line	Circle	Any	Trochoid
Line	Circle	Point on the circle	Cycloid
Line	Conic section	Center of the conic	Sturm roulette^[2]
Line	Conic section	Focus of the conic	Delaunay roulette^[3]
Line	Parabola	Focus of the parabola	Catenary^[4]
Line	Ellipse	Focus of the ellipse	Elliptic catenary^[4]
Line	Hyperbola	Focus of the hyperbola	Hyperbolic catenary^[4]
Line	Epicycloid or Hypocycloid	Center	Ellipse^[5]
Parabola	Equal parabola parameterized in opposite direction	Vertex of the parabola	Cissoid of Diocles^[1]
Catenary	Line	See example	Line