Given a curve and a fixed point called the pedal point, then for a point on , draw a line perpendicular to . The envelope of these lines as describes the curve is the negative pedal of . 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 twopoint 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. 4748).
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  ovoid 
ellipse negative pedal curve with  focus  twocusped curve 
line  any point  parabola 
parabola negative pedal curve  origin  semicubical parabola 
parabola negative pedal curve  focus  Tschirnhausen cubic 