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.