The polar curve
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
It has area
and arc length
(OEIS 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
See alsoAngle Trisection
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ReferencesLoomis, 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,
Cite this as:
Weisstein, Eric W. "Cycloid of Ceva."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CycloidofCeva.html