Negative Pedal Curve

The negative pedal curve of a curve with respect to a fixed pedal point is the envelope of the lines through points of and perpendicular to . It can be constructed by considering the perpendicular line segment for a curve parameterized by . Since one end of the perpendicular corresponds to the point , . Another end point can be found by taking the perpendicular to the line, giving

(1)

or

			(2)
			(3)

Plugging into the two-point form of a line then gives

(4)

or

(5)

Solving the simultaneous equations and then gives the equations of the negative pedal curve as

			(6)
			(7)

If a curve is the pedal curve of a curve , then is the negative pedal curve of (Lawrence 1972, pp. 47-48).

The focal negative pedal curves of conic sections form the ichthyoid curve family (Cundy 2001).

The following table summarizes the negative pedal curves for some common curves.

curve	pedal point	negative pedal curve
cardioid negative pedal curve	origin	circle
cardioid negative pedal curve	point opposite cusp	cissoid of Diocles
circle negative pedal curve	inside the circle	ellipse
circle negative pedal curve	outside the circle	hyperbola
ellipse negative pedal curve with	center	Talbot's curve
ellipse negative pedal curve with	focus	Burleigh's oval
ellipse negative pedal curve with	focus	two-cusped Burleigh's oval
ellipse negative pedal curve with	focus	fish curve
conic section	focus	ichthyoid curve
line	any point	parabola
parabola negative pedal curve	origin	semicubical parabola
parabola negative pedal curve	focus	Tschirnhausen cubic

Burleigh's oval is convex for , while gives its singular transition member. The fish curve is the rectangular ichthyoid with .

The negative pedal curve of the arc of a circle involute, with respect to the center of its generating circle, is an Atzema spiral up to a rotation.