Right Strophoid

A right strophoid is the strophoid of a line with pole not on and fixed point being the point where the perpendicular from to cuts is called a right strophoid. It is therefore a general strophoid with .

The right strophoid is given by the Cartesian equation

(1)

or the polar equation

(2)

The parametric form of the strophoid is

			(3)
			(4)

The right strophoid has curvature, arc length, and tangential angle given by

			(5)
			(6)
			(7)

where

			(8)
			(9)

, and are incomplete elliptic integrals of the first, second, and third kinds, respectively.

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, corresponding to , is given by

			(10)
			(11)
			(12)
			(13)

(MacTutor Archive). The arc length of the loop is given by

(14)

where is again defined as above.

Let 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 and is therefore an anallagmatic curve.