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Cycloid of Ceva

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CycloidofCeva

The cycloid of Ceva is the polar curve

r=a[1+2cos(2theta)]
(1)

for a>0 that can be used for angle trisection. Loomis (1968, p. 29) attributes the curve to Ceva in 1699, under the name cycloidum anomalarum. It has Cartesian equation

(x^2+y^2)^3=a^2(3x^2-y^2)^2.
(2)

It has area

A=3pia^2
(3)

and arc length

s=a[16E(k)-3K(k)+3Pi(1/4|k^2)]
(4)
approx 20.01578...a
(5)

(OEIS A138497), with k=sqrt(13)/4, where K(k), E(k), and Pi(n|m) are complete elliptic integrals of the first, second, and third, respectively. Here K and E use the elliptic modulus k, while Pi uses the elliptic parameter m=k^2 and characteristic n=1/4.

The arc length function can be expressed in closed form in terms of elliptic integrals, and the curvature at polar angle t=theta is

kappa(t)=(3[9+4cos(2t)-2cos(4t)])/(a[11+4cos(2t)-6cos(4t)]^(3/2)).
(6)

See also

Angle Trisection, Cycloid, Trisectrix

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References

Loomis, E. S. "The Cycloid of Ceva." §2.7 in The Pythagorean Proposition: Its Demonstrations Analyzed and Classified and Bibliography of Sources for Data of the Four Kinds of "Proofs," 2nd ed. Reston, VA: National Council of Teachers of Mathematics, pp. 29-30, 1968.Loy, J. "Trisection of an Angle." https://web.archive.org/web/20030402133520/http://www.jimloy.com/geometry/trisect.htm#curves.Sloane, N. J. A. Sequence A138497 in "The On-Line Encyclopedia of Integer Sequences."Yates, R. C. The Trisection Problem. Reston, VA: National Council of Teachers of Mathematics, 1971.

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Cycloid of Ceva

Cite this as:

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

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