The polar curve

(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

(2)

It has area

(3)

and arc length

			(4)
			(5)

(Sloane's A138497), with , where , , and 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

(6)

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. http://mathworld.wolfram.com/CycloidofCeva.html