Cycloid of Ceva
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
![画像: kappa(t)=(3[9+4cos(2t)-2cos(4t)])/([11+4cos(2t)-6cos(4t)]^(3/2)).](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/CycloidofCeva/NumberedEquation4.svg){kind=link}
See also
Angle Trisection, Cycloid, TrisectrixExplore with Wolfram|Alpha
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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.Referenced on Wolfram|Alpha
Cycloid of CevaCite this as:
Weisstein, Eric W. "Cycloid of Ceva." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CycloidofCeva.html