Cayley's Sextic

$\begin{figure}\begin{center}\BoxedEPSF{cayleys_sextic.epsf scaled 700}\end{center}\end{figure}$

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

$\begin{displaymath} r=a\cos^3({\textstyle{1\over 3}}\theta), \end{displaymath}$

(1)

$\begin{displaymath} r=4b\cos^3({\textstyle{1\over 3}}\theta), \end{displaymath}$

(2)

where $b\equiv a/4$ . In the latter case, the Cartesian equation is

$\begin{displaymath} 4(x^2+y^2-bx)^3=27a^2(x^2+y^2)^2. \end{displaymath}$

(3)

The parametric equations are

$\displaystyle x(t)$	$\textstyle =$	$\displaystyle 4a\cos^4({\textstyle{1\over 2}}t)(2\cos t-1)$	(4)
$\displaystyle y(t)$	$\textstyle =$	$\displaystyle 4a\cos^3({\textstyle{1\over 2}}t)\sin({\textstyle{3\over 2}}t).$	(5)

$\begin{figure}\begin{center}\BoxedEPSF{CayleysSexticInfo.epsf scaled 700}\end{center}\end{figure}$

The Arc Length, Curvature, and Tangential Angle are

$\displaystyle s(t)$	$\textstyle =$	$\displaystyle 3(t+\sin t),$	(6)
$\displaystyle \kappa(t)$	$\textstyle =$	$\displaystyle {\textstyle{1\over 3}}\sec^2({\textstyle{1\over 2}}t),$	(7)
$\displaystyle \phi(t)$	$\textstyle =$	$\displaystyle 2t.$	(8)

References

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.