Roulette (curve)

In the differential geometry of curves, a roulette is a kind of curve, generalizing cycloids, epicycloids, hypocycloids, trochoids, epitrochoids, hypotrochoids, and involutes.

DefinitionEdit

Informal definitionEdit

 
A green parabola rolls along an equal blue parabola which remains fixed. The generator is the vertex of the rolling parabola and describes the roulette, shown in red. In this case the roulette is the cissoid of Diocles.[1]

Roughly speaking, a roulette is the curve described by a point (called the generator or pole) attached to a given curve as that curve 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.

Special cases and related conceptsEdit

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.

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.

Formal definitionEdit

Formally speaking, the curves must be differentiable curves in the Euclidean plane. The fixed curve is kept invariant; the rolling curve 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 (another way to express this constraint is that the point of contact of the two curves is the instant centre of rotation of the congruence transformation). The resulting roulette is formed by the locus of the generator subjected to the same set of congruence transformations.

Modeling the original curves as curves in the complex plane, let   be the two natural parameterizations of the rolling ( ) and fixed ( ) curves, such that  ,  , and   for all  . The roulette of generator   as   is rolled on   is then given by the mapping:

 

GeneralizationsEdit

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 a roulette.

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

ExampleEdit

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

 
 

The parameterization of the line is chosen so that

 

Applying the formula above we obtain:

 

If p = −i the expression has a constant imaginary part (namely −i) and the roulette is a horizontal line. An interesting application of this is that a square wheel could roll without bouncing on a road that is a matched series of catenary arcs.

List of roulettesEdit

Fixed curve Rolling curve Generating point Roulette
Any curve Line Point on the line Involute of the curve
Line Any Any Cyclogon
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 Hyperbola Center of the hyperbola Rectangular elastica[2][failed verification]
Line Cyclocycloid Center Ellipse[5]
Circle Circle Any Centered trochoid[6]
Outside of a circle Circle Any Epitrochoid
Outside of a circle Circle Point on the circle Epicycloid
Outside of a circle Circle of identical radius Any Limaçon
Outside of a circle Circle of identical radius Point on the circle Cardioid
Outside of a circle Circle of half the radius Point on the circle Nephroid
Inside of a circle Circle Any Hypotrochoid
Inside of a circle Circle Point on the circle Hypocycloid
Inside of a circle Circle of a third of the radius Point on the circle Deltoid
Inside of a circle Circle of a quarter of the radius Point on the circle Astroid
Parabola Equal parabola parameterized in opposite direction Vertex of the parabola Cissoid of Diocles[1]
Catenary Line See example above Line

See alsoEdit

NotesEdit

ReferencesEdit

  • W. H. Besant (1890) Notes on Roulettes and Glissettes from Cornell University Historical Math Monographs, originally published by Deighton, Bell & Co.
  • Weisstein, Eric W. "Roulette". MathWorld.

Further readingEdit