The catacaustic of a parabola 
opening upward is complicated
for a general radiant point 
. However, the equations simplify substantially in the
case 
(i.e., with rays perpendicular to the axis of the
parabola), giving
 |  |  |
(1)
|
 |  |  |
(2)
|
Making the substitution 
yields
 |  |  |
(3)
|
 |  |  |
(4)
|
which is a translated and rotated Tschirnhausen cubic with 
.
If the radiant point is taken at 
(i.e., with rays parallel
to the axis of the parabola), then the catacaustic
degenerates to the single point 
as expected since the parabola
has a focus at this point.
