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.

Curve 1	Curve 2	Pole	Roulette
Circle	exterior Circle	on Circumference	Epicycloid
Circle	interior Circle	on Circumference	Hypocycloid
Circle	Line	on Circumference	Cycloid
Circle	same Circle	any point	Rose
Circle Involute	Line	Center	Parabola
Cycloid	Line	center	Ellipse
Ellipse	Line	Focus	elliptic catenary
Hyperbola	Line	Focus	hyperbolic catenary
Hyperbolic Spiral	Line	Origin	Tractrix
Line	any curve	on Line	Involute of the curve
Logarithmic Spiral	Line	any point	Line
Parabola	equal Parabola	Vertex	Cissoid of Diocles
Parabola	Line	Focus	Catenary

See also Glissette, Unduloid

References

Besant, W. H. Notes on Roulettes and Glissettes, 2nd enl. ed. Cambridge, England: Deighton, Bell & Co., 1890.

Cundy, H. and Rollett, A. ``Roulettes and Involutes.'' §2.6 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 46-55, 1989.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 56-58 and 206, 1972.

Lockwood, E. H. ``Roulettes.'' Ch. 17 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 138-151, 1967.

Yates, R. C. ``Roulettes.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 175-185, 1952.

Zwillinger, D. (Ed.). ``Roulettes (Spirograph Curves).'' §8.2 in CRC Standard Mathematical Tables and Formulae, 3rd ed. Boca Raton, FL: CRC Press, 1996. http://www.geom.umn.edu/docs/reference/CRC-formulas/node34.html.