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.