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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


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.

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© 1996-9 Eric W. Weisstein