reflecting wall curve

with the following f_a(t):

Suppose throwing a ball from a fixed point, say the origin, at a given angle φ, with varying initial velocity. Then an orthogonal trajectory of the parabolic paths can be seen as the wall that would bounce (reflect) the ball back to the origin. We call this trajectory the reflecting wall curve.
The tangent of the angle φ corresponds with the value a in the formulas above.

Darryl Nester worked on this curve, he constructed an animated gif to show the curve as function of the angle.