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Cochleoid

\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,

\begin{displaymath}
r={a\sin\theta\over\theta}.
\end{displaymath} (1)

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

The Curvature is
\begin{displaymath}
\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


References

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

MacTutor History of Mathematics Archive. ``Cochleoid.'' http://www-groups.dcs.st-and.ac.uk/~history/Curves/Cochleoid.html.




© 1996-9 Eric W. Weisstein
1999-05-26