Right Strophoid

The Strophoid of a line $L$ with pole $O$ not on $L$ and fixed point $O'$ being the point where the Perpendicular from $O$ to $L$ cuts $L$ is called a right strophoid. It is therefore a general Strophoid with $a=\pi/2$.

The right strophoid is given by the Cartesian equation

$$y^2={c-x\over c+x} x^2,$$ (1)

or the polar equation

$$r=c\cos(2\theta)\sec\theta.$$ (2)

The parametric form of the strophoid is

$$x(t) = {1-t^2\over t^2+1}$$ (3)

$$y(t) = {t(t^2-1)\over t^2+1}.$$ (4)

The right strophoid has Curvature

$$\kappa(t)=-{4(1+3t^2)\over(1+6t^2+t^4)^{3/2}}$$ (5)

and Tangential Angle

$$\phi(t)=-2\tan^{-1} t-\tan^{-1}\left({2t\over 1+t^2}\right).$$ (6)

The right strophoid first appears in work by Isaac Barrow in 1670, although Torricelli describes the curve in his letters around 1645 and Roberval found it as the Locus of the focus of the conic obtained when the plane cutting the Cone rotates about the tangent at its vertex (MacTutor Archive). The Area of the loop is

$$A_{\rm loop}={\textstyle{1\over 2}}c^2(4-\pi)$$ (7)

(MacTutor Archive).

Let $C$ be the Circle with center at the point where the right strophoid crosses the $x$-axis and radius the distance of that point from the origin. Then the right strophoid is invariant under inversion in the Circle $C$ and is therefore an Anallagmatic Curve.

See also Strophoid, Trisectrix

References

Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, p. 71, 1993.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 100-104, 1972.

Lockwood, E. H. ``The Right Strophoid.'' Ch. 10 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 90-97, 1967.

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