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\begin{figure}\begin{center}\BoxedEPSF{cochleoid.epsf scaled 800}\end{center}\end{figure}

The cochleoid, whose name means ``snail-form'' in Latin, was first discussed by J. Peck in 1700 (MacTutor Archive). The points of contact of Parallel Tangents to the cochleoid lie on a Strophoid.

In Polar Coordinates,

\end{displaymath} (1)

In Cartesian Coordinates,
(x^2+y^2)\tan^{-1}\left({y\over x}\right)=ay.
\end{displaymath} (2)

The Curvature is
\kappa={2\sqrt{2}\,\theta^3[2\theta-\sin(2\theta)]\over [1+2\theta^2-\cos(2\theta)-2\theta\sin(2\theta)]^{3/2}}.
\end{displaymath} (3)

See also Quadratrix of Hippias


Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 192 and 196, 1972.

MacTutor History of Mathematics Archive. ``Cochleoid.''

© 1996-9 Eric W. Weisstein