TOPICS
Search

Cycloid


Cycloid
CycloidFrames
Cycloid

The cycloid is the locus of a point on the rim of a circle of radius a rolling along a straight line. It was studied and named by Galileo in 1599. Galileo attempted to find the area by weighing pieces of metal cut into the shape of the cycloid. Torricelli, Fermat, and Descartes all found the area. The cycloid was also studied by Roberval in 1634, Wren in 1658, Huygens in 1673, and Johann Bernoulli in 1696. Roberval and Wren found the arc length (MacTutor Archive). Gear teeth were also made out of cycloids, as first proposed by Desargues in the 1630s (Cundy and Rollett 1989).

In 1696, Johann Bernoulli challenged other mathematicians to find the curve which solves the brachistochrone problem, knowing the solution to be a cycloid. Leibniz, Newton, Jakob Bernoulli and L'Hospital all solved Bernoulli's challenge. The cycloid also solves the tautochrone problem, as alluded to in the following passage from Moby Dick: "[The try-pot] is also a place for profound mathematical meditation. It was in the left-hand try-pot of the Pequod, with the soapstone diligently circling round me, that I was first indirectly struck by the remarkable fact, that in geometry all bodies gliding along a cycloid, my soapstone, for example, will descend from any point in precisely the same time" (Melville 1851). Because of the frequency with which it provoked quarrels among mathematicians in the 17th century, the cycloid became known as the "Helen of Geometers" (Boyer 1968, p. 389).

The cycloid catacaustic when the rays are parallel to the y-axis is a cycloid with twice as many arches. The radial curve of a cycloid is a circle. The evolute and involute of a cycloid are identical cycloids.

If the cycloid has a cusp at the origin and its humps are oriented upward, its parametric equation is

x=a(t-sint)
(1)
y=a(1-cost).
(2)

Humps are completed at t values corresponding to successive multiples of 2pi, and have height 2a and length 2pia. Eliminating t in the above equations gives the Cartesian equation

 x=acos^(-1)(1-y/a)-sqrt(2ay-y^2)
(3)

which is valid for y in [0,2a] and gives the first half of the first hump of the cycloid. An implicit Cartesian equation is given by

 |x/a+2pi[1/2-2/(2pi)x/a]-1|=cos^(-1)(1-y/a)-sqrt(2y/a-(y/a)^2).
(4)

The arc length, curvature, and tangential angle for the first hump of the cycloid are

s(t)=4a{(-1)^(|_t/(2pi)+1/2_|)[1-|cos(1/2t)|]|sin(1/2t)|csc(1/2t)+2|_t/(2pi)+1/2_|}
(5)
kappa(t)=-(|csc(1/2t)|)/(4a)
(6)
phi(t)=1/2(pi-t+2pi|_t/(2pi)_|).
(7)

For the first hump,

 s(t)=8asin^2(1/4t).
(8)

For a single hump of the cycloid, the arc length and area under the curve are therefore

L=8a
(9)
A=3pia^2.
(10)

See also

Brachistochrone Problem, Curtate Cycloid, Cyclide, Cycloid Catacaustic, Cycloid Evolute, Cycloid Involute, Epicycloid, Hypocycloid, Prolate Cycloid, Tautochrone Problem, Trochoid

Explore with Wolfram|Alpha

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 216, 1987.Bogomolny, A. "Cycloids." http://www.cut-the-knot.org/pythagoras/cycloids.shtml.Boyer, C. B. A History of Mathematics. New York: Wiley, 1968.Cundy, H. and Rollett, A. "Cycloid." §5.1.6 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 215-216, 1989.Gardner, M. "The Cycloid: Helen of Geometers." Ch. 13 in The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 127-134, 1984.Gray, A. "Cycloids." §3.1 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 50-52, 1997.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 197, 1972.Lockwood, E. H. "The Cycloid." Ch. 9 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 80-89, 1967.MacTutor History of Mathematics Archive. "Cycloid." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Cycloid.html.Melville, H. "The Tryworks." Ch. 96 in Moby Dick. New York: Bantam, 1981. Originally published in 1851.Update a linkMuterspaugh, J.; Driver, T.; and Dick, J. E. "The Cycloid and Tautochronism." http://php.indiana.edu/~jedick/project/intro.htmlPappas, T. "The Cycloid--The Helen of Geometry." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 6-8, 1989.Phillips, J. P. "Brachistochrone, Tautochrone, Cycloid--Apple of Discord." Math. Teacher 60, 506-508, 1967.Proctor, R. A. A Treatise on the Cycloid. London: Longmans, Green, 1878.Smith, D. E. History of Mathematics, Vol. 2: Special Topics of Elementary Mathematics. New York: Dover, p. 327, 1958.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 147, 1999.Wagon, S. "Rolling Circles." Ch. 2 in Mathematica in Action. New York: W. H. Freeman, pp. 39-66, 1991.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 44-47, 1991.Whitman, E. A. "Some Historical Notes on the Cycloid." Amer. Math. Monthly 50, 309-315, 1948.Yates, R. C. "Cycloid." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 65-70, 1952.Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 291-292, 1995.

Cite this as:

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

Subject classifications