## 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

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.