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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

(1)

The Cartesian equation is

(2)

Parametric equations can be given by

			(3)
			(4)

for . In this parametrization, the loop corresponds to .

The area enclosed by the outer boundary is

			(5)
			(6)

(Sloane's A118308), and by the inner loop is

			(7)
			(8)

(Sloane's A118309), and the arc length of the entire curve is

(9)

The arc length, curvature, and tangential angle are given by

			(10)
			(11)
			(12)

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." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Cayleys.html.

Sloane, N. J. A. Sequences A118308 and A118309 in "The On-Line Encyclopedia of Integer Sequences."

CITE THIS AS:

Weisstein, Eric W. "Cayley's Sextic." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CayleysSextic.html