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Let $C$ be a curve, let $O$ be a fixed point (the Pole), and let $O'$ be a second fixed point. Let $P$ and $P'$ be points on a line through $O$ meeting $C$ at $Q$ such that $P'Q = QP = QO'$. The Locus of $P$ and $P'$ is called the strophoid of $C$ with respect to the Pole $O$ and fixed point $O'$. Let $C$ be represented parametrically by $(f(t),g(t))$, and let $O=(x_0,y_0)$ and $O'=(x_1,y_1)$. Then the equation of the strophoid is

$\displaystyle x$ $\textstyle =$ $\displaystyle f\pm\sqrt{(x_1-f)^2+(y_1-g)^2\over 1+m^2}$ (1)
$\displaystyle y$ $\textstyle =$ $\displaystyle g\pm\sqrt{(x_1-f)^2+(y_1-g)^2\over 1+m^2},$ (2)

m\equiv {g-y_0\over f-x_0}.
\end{displaymath} (3)

The name strophoid means ``belt with a twist,'' and was proposed by Montucci in 1846 (MacTutor Archive). The polar form for a general strophoid is
r = {b\sin(a-2\theta)\over \sin(a-\theta)}.
\end{displaymath} (4)

If $a=\pi/2$, the curve is a Right Strophoid. The following table gives the strophoids of some common curves.

Curve Pole Fixed Point Strophoid
line not on line on line oblique strophoid
line not on line foot of Perpendicular origin to line Right Strophoid
Circle center on the circumference Freeth's Nephroid

See also Right Strophoid


Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 51-53 and 205, 1972.

Lockwood, E. H. ``Strophoids.'' Ch. 16 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 134-137, 1967.

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

Yates, R. C. ``Strophoid.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 217-220, 1952.

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© 1996-9 Eric W. Weisstein