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Roulette


A roulette is a 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.

Rolling 3-gon
Rolling 4-gon
Rolling 5-gon
Rolling 6-gon

A particularly interesting case of a roulette is a regular n-gon rolling on a "road" composed of a sequence of truncated catenaries, as illustrated above, as first noted by Robison (1960). 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 (a "square wheel"), the shape of the road is the catenary y=-coshx truncated at x=+/-sinh^(-1)1 (Wagon 2000). Interestingly, a detailed mathematical analysis had been published by J. C. Maxwell (of electromagnetic theory fame) in 1849 (Maxwell 1849) which, while its did not include critical idea of truncating catenaries, contained essentially all the other underlying mathematical ideas.

For a regular n-gon, the Cartesian equation of the corresponding catenary is

 y=-Acosh(x/A),

where

 A=Rcot(pi/n).

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 curverolling curvepoleroulette
any curvelineon lineinvolute of the curve
circleexterior circle of radius ab<acurtate epicycloid
circleexterior circle of radius ab=aepicycloid
circleexterior circle of radius ab>aprolate epicycloid
circleinterior circle of radius ab<acurtate hypocycloid
circleinterior circle of radius ab=ahypocycloid
circleinterior circle of radius ab>aprolate hypocycloid
circleexterior equal circleany pointrose curve
linecircle of radius ab<acurtate cycloid
linecircle of radius ab=acycloid
linecircle of radius ab>aprolate cycloid
linecircle involutecenterparabola
linecycloidcenterellipse
lineellipsefocuselliptic catenary
linehyperbolafocushyperbolic catenary
linehyperbolic spiralorigintractrix
linelogarithmic spiralany pointline
lineparabolafocuscatenary
parabolaequal parabolaparabola vertexcissoid of Diocles

See also

Catenary, Delta Curve, Glissette, Reuleaux Polygon, Reuleaux Triangle, Rotor, Unduloid

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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.Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, p. 128, 1984.Hall, L. and Wagon, S. "Roads and Wheels." Math. Mag. 65, 283-301, 1992.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.Maxwell, J. C. "On the Theory of Rolling Curves." §XXXV in Trans. Roy. Soc. Edin. 16, 519-540, 1849.Robison, G. B. "Rockers and Rollers." Math. Mag. 33, 139-144, 1960.Wasgon, S. "The Ultimate Flat Tire." Math. Horizons, pp. 14-17, Feb. 1999.Wagon, S. Mathematica in Action, 2nd ed. New York: W. H. Freeman, p. 52, 2000.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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Roulette

Cite this as:

Weisstein, Eric W. "Roulette." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Roulette.html

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