Freeth's Nephroid

Freeth's nephroid is a strophoid of a circle with the pole at the center of the circle and the fixed point on the circumference of the circle. In the construction illustrated above, is a moving point on the circle, and and lie on the line in opposite directions from . The point is obtained by moving away from , and by moving toward , in each case through a distance . Thus , as indicated by the matching marks. As moves around the circle, the two points together trace Freeth's nephroid. Freeth (1878, pp. 130 and 228) described this and various other strophoids (MacTutor Archive).

The perpendicular bisector of cuts the outer boundary of Freeth's nephroid at the upper point illustrated above. The angle is , so this curve can be used to construct a regular heptagon.

Freeth's nephroid has polar equation

(1)

for .

The area enclosed by the outer boundary of the curve is

(2)

and the total arc length over is

			(3)
			(4)

(OEIS A138498), where , is a complete elliptic integral of the first kind, is a complete elliptic integral of the second kind, and is a complete elliptic integral of the third kind.

For , the curvature and the change in tangential angle from its initial value are given by

			(5)
			(6)

where is the floor function. Here , so the absolute tangential angle is . At odd multiples of , the value of is defined by the limit of the expression above.