Roulette

The curve traced by a fixed point on a closed convex curve as that curve rolls without slipping along a second curve. The roulettes described by the foci of conics when rolled upon a line are sections of minimal surfaces (i.e., they yield minimal surfaces when revolved about the line) known as unduloids.

A particularly interesting case of a roulette is a regular -gon rolling on a "road" composed of a sequence of truncated catenaries, as illustrated above. 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, the shape of the road is the catenary truncated at (Wagon 2000). For a regular -gon, the Cartesian equation of the corresponding catenary is

where

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
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

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