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.
© 1996-9 Eric W. Weisstein