Circle Pedal Curve

The circle pedal curve for a unit circle with parametric equation

			(1)
			(2)

with pedal point is

			(3)
			(4)

The pedal curve with respect to the center is the circle itself (Gray 1997, pp. 119 and 124-135).

If the pedal point is taken on the circumference (and in particular at the point ), the pedal curve is the cardioid

			(5)
			(6)

and otherwise is a limaçon.

More generally, let the circle have radius , and let be the distance from its center to the pedal point. After a rotation of coordinates, the squared speed of the pedal curve is

(7)

It follows that the total arc length is

(8)

where is the complete elliptic integral of the second kind with modulus . The signed area of the standard full traversal is

(9)