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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

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

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
\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)

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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)


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.''

© 1996-9 Eric W. Weisstein