Cayley's Sextic
A plane curve discovered by Maclaurin but first studied in detail by Cayley. The name Cayley's sextic is due to R. C. Archibald, who attempted to classify curves in a paper published in Strasbourg in 1900 (MacTutor Archive). Cayley's sextic is given in polar coordinates by
| r=4acos^3(1/3theta). |
(1)
|
The Cartesian equation is
Parametric equations can be given by
for 0<t<3pi. In this parametrization, the loop corresponds to pi<t<2pi.
The area enclosed by the outer boundary is
(OEIS A118308), and by the inner loop is
(OEIS A118309), and the arc length of the entire curve is
| s=6pia. |
(9)
|
The arc length, curvature, and tangential angle are given by
See also
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References
Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 119-120, 1997.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 178 and 180, 1972.MacTutor History of Mathematics Archive. "Cayley's Sextic." https://mathshistory.st-andrews.ac.uk/Curves/Cayleys/.Sloane, N. J. A. Sequences A118308 and A118309 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Cayley's SexticCite this as:
Weisstein, Eric W. "Cayley's Sextic." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CayleysSextic.html