TOPICS
Search

Cycloid of Ceva

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook
CycloidofCeva

The polar curve

r=1+2cos(2theta)
(1)

that can be used for angle trisection. It was devised by Ceva in 1699, who termed it the cycloidum anomalarum (Loomis 1968, p. 29). It has Cartesian equation

(x^2+y^2)^3=(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)]
(4)
=20.01578...a
(5)

(OEIS A138497), with k=sqrt(13)/4, where K(k), E(k), and Pi(z,k) are complete elliptic integrals of the first, second, and third, respectively.

The arc length function is a slightly complicated expression that can be expressed in closed form in terms of elliptic functions, and the curvature is given by

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

See also

Angle Trisection, Cycloid, Trisectrix

Explore with Wolfram|Alpha

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." 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.

Cite this as:

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

Subject classifications