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Prolate Cycloid

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ProlateCycloid
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Prolate Cycloid

The path traced out by a fixed point at a radius b>a, where a is the radius of a rolling circle, also sometimes called an extended cycloid. The prolate cycloid contains loops, and has parametric equations

x=aphi-bsinphi
(1)
y=a-bcosphi.
(2)

The arc length from phi=0 is

s=2(a+b)E(u),
(3)

where

sin(1/2phi)=snu
(4)
k^2=(4ab)/((a+c)^2).
(5)

See also

Curtate Cycloid, Cycloid, Prolate Cycloid Evolute, Trochoid

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References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 216, 1987.Harris, J. W. and Stocker, H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 325, 1998.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 192 and 194-197, 1972.Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 146, 1967.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 147-148, 1999.Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 292, 1995.

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Prolate Cycloid

Cite this as:

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

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