Strophoid

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

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

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

where

$$m\equiv {g-y_0\over f-x_0}.$$ (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

$$r = {b\sin(a-2\theta)\over \sin(a-\theta)}.$$ (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

References

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.'' http://www-groups.dcs.st-and.ac.uk/~history/Curves/Right.html.

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