Roulette

A roulette is a curve traced by a fixed point rigidly attached to a plane curve as that curve rolls without slipping along a second curve. The roulettes traced by the foci of conics rolling on a line generate surfaces of revolution with constant mean curvature when revolved about that line (Delaunay 1841, p. 315). A noncircular ellipse gives an unduloid, while a parabola gives a catenoid.

All four regular polygons have side length 2, and their geometric centroids advance with the same constant horizontal velocity.

A particularly interesting case of a roulette is a regular n-gon rolling on a "road" composed of a sequence of truncated catenaries, as illustrated above, as first noted by Robison (1960). This motion is smooth in the sense that the geometric centroid follows a straight line, although in the case of the rolling equilateral triangle, a physical model would be impossible to construct because the vertices of the triangles would get "stuck" in the ruts (Wagon 2000). For the rolling square (a "square wheel"), the shape of the road is the catenary truncated at (Wagon 2000). Maxwell (1849) had already studied the rolling line and catenary.

For a regular n-gon, the Cartesian equation of the corresponding catenary is

(1)

with

(2)

where each side has length .

Slavík and Wagon (2026, pp. 1-3) derive the wheel-road relation from no slipping. At constant nonzero angular velocity, this means that each point on the wheel has zero instantaneous velocity when it contacts the road. Consider a wheel with marked center initially at the origin, moving horizontally while the wheel rotates clockwise. Write its initial shape as

(3)

where is a continuous function on an interval containing 0 and is the initial contact point. The corresponding road has parametric equations

			(4)
			(5)

with also equal to the clockwise rotation angle. The marked center remains directly above the contact point. For a wheel initially shaped as the line , taking on gives and recovers the inverted catenary . Restricting to gives one arch of the road for a square of side length 2.

This construction does not require to be differentiable, so it also applies when is a nowhere differentiable function. It does not by itself exclude collisions between other parts of the wheel and the road. When is piecewise continuously differentiable, the arc lengths of wheel and road between corresponding contact points are equal as a consequence of no slipping. Equality of these arc lengths alone is not sufficient to define rolling (Slavík and Wagon 2026, pp. 3-4, 9-10).

The roulette consisting of a square on a truncated catenary road is depicted on the cover of Wagon (2000).

Given a base curve, let another curve roll on it, and call the point rigidly attached to this rolling curve the "pole." The following table then summarizes some roulettes for various common curves and poles. Note that the cases curtate cycloid, cycloid, and prolate cycloid are together called trochoids, and similarly for the various varieties of epicycloids (called epitrochoids) and hypocycloids (called hypotrochoids).

base curve	rolling curve	pole	roulette
any curve	line	on line	involute of the curve
circle	exterior circle of radius		curtate epicycloid
circle	exterior circle of radius		epicycloid
circle	exterior circle of radius		prolate epicycloid
circle	interior circle of radius		curtate hypocycloid
circle	interior circle of radius		hypocycloid
circle	interior circle of radius		prolate hypocycloid
circle	exterior equal circle	any point	rose curve
line	circle of radius		curtate cycloid
line	circle of radius		cycloid
line	circle of radius		prolate cycloid
line	circle involute	center	parabola
line	cycloid	center	ellipse
line	ellipse	focus	elliptic catenary
line	hyperbola	focus	hyperbolic catenary
line	hyperbolic spiral	origin	tractrix
line	logarithmic spiral	any point	line
line	parabola	focus	catenary
parabola	equal parabola	parabola vertex	cissoid of Diocles