 |  |  |
 |  |
The pedal curve of the parabola with parametric equations
 |  |  |
(1)
|
 |  |  |
(2)
|
with pedal point 
is
 |  |  |
(3)
|
 |  |  |
(4)
|
On the conic section directrix, the pedal curve of a parabola
is a strophoid (top left). On the foot of the conic
section directrix, it is a right strophoid
(top middle). On reflection of the focus in the conic
section directrix, it is a Maclaurin trisectrix
(top right). On the parabola vertex, it is a cissoid of Diocles (bottom left; Gray 1997, p. 119).
On the focus, it is a straight line (bottom right; Hilbert
and Cohn-Vossen 1999, pp. 26-27). On the symmetry axis for a parabola with 
, it is a conchoid
of de Sluze (H. Smith, pers. comm., Aug. 4, 2004). The following table
summarizes these special cases.
| pedal point | pedal curve |
| directrix | strophoid |
| foot of directrix | right strophoid |
| reflection of focus in directrix | Maclaurin trisectrix |
| parabola vertex | cissoid of Diocles |
| focus | line |
axis
of a parabola with  | conchoid of de Sluze |
