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 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
 |  | ![-([(g-y_0)^2-(f-x_0)f]g^'+(2f-x_0)(g-y_0)f^')/((f-x_0)g^'-(g-y_0)f^')](/images/equations/NegativePedalCurve/Inline25.svg) |
(6)
|
 |  | ![([(f-x_0)^2-(g-y_0)g]f^'+(2g-y_0)(f-x_0)g^')/((f-x_0)g^'-(g-y_0)f^').](/images/equations/NegativePedalCurve/Inline28.svg) |
(7)
|
If a curve 
is the pedal curve of a curve 
, then 
is the negative pedal curve of 
(Lawrence 1972, pp. 47-48).
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 | two-cusped curve |
| line | any point | parabola |
| parabola negative pedal curve | origin | semicubical parabola |
| parabola negative pedal curve | focus | Tschirnhausen cubic |
