Art wrote:
"I thought you were getting bored with physics!"

We must live with physics, bored or not. The parabola is well
understood. its use as a reflector is documented by Kraus and the "ARRL
Antenna Book". A good description is seen in "Principles of Radar"
published by the MIT Radar School Staff in 1946. On page 9-78:

"The geometrical properties of parabolas are important for demonstrating
the existence of a constant-phase surface. First, a parabola is by
definition the focus of points as far from a fixed point called the
focus as from a fixed line called the directrix. With reference to Fig.
51A, this means that lengths AA` and AF are equal, BB`and BF are equal,
and so on. Second, a line drawn tangent to a parabola at any point (as
in Fig. 51B) makes equal angles with a line drawn from this point back
to the focus and a line from this point parallel to the axis of the
parabola. When a point source is placed at the focus, it sends out
energy in a single time phase, but in various directions. This energy
strikes the paraboloid at points such as A, B, and C, (in Fig.51A), and
is reflected in a direction parallel with the axis because of the second
property mentioned. The first property predicts that the phase change
that the wave undergoes in traveling to points A``, B``, and C`` on the
surface SS` is the same for each ray, the phase change being equal to
the distance in electrical degrees, from the diretrix to the surface SS`
plus 180 degrees, due to the phase reversal upon reflection. Thus the
field reflected from the parabola has a single time phase in a plane
across the mouth of the parabola. The field radiated forward by the
point source tends to upset this constant-phase surface, but this effect
is usually minimized through the use of sources which radiate
appreciably only roward the reflector.

Best regards, Richard Harrison, KB5WZI