Conchoid

A curve whose name means ``shell form.'' Let $C$ be a curve and $O$ a fixed point. Let $P$ and $P'$ be points on a line from $O$ to $C$ meeting it at $Q$, where $P'Q = QP = k$, with $k$ a given constant. For example, if $C$ is a Circle and $O$ is on $C$, then the conchoid is a Limaçon, while in the special case that $k$ is the Diameter of $C$, then the conchoid is a Cardioid. The equation for a parametrically represented curve $(f(t),g(t))$ with $O=(x_0,y_0)$ is

$$\begin{eqnarray*} x&=&f\pm {k(f-x_0)\over\sqrt{(f-x_0)^2+(g-y_0)^2}}\\ y&=&g\pm {k(g-y_0)\over\sqrt{(f-x_0)^2+(g-y_0)^2}}. \end{eqnarray*}$$

See also Concho-Spiral, Conchoid of de Sluze, Conchoid of Nicomedes, Conical Spiral, Dürer's Conchoid

References

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 49-51, 1972.

Lee, X. ``Conchoid.'' http://www.best.com/~xah/SpecialPlaneCurves_dir/Conchoid_dir/conchoid.html.

Lockwood, E. H. ``Conchoids.'' Ch. 14 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 126-129, 1967.

Yates, R. C. ``Conchoid.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 31-33, 1952.