The Geometry of the Foucault Test

A parabola is flatter on the outer edges, so reflections from there will focus farther away from the mirror.
So how far away from the mirror does each zone on the mirror focus? First, let's look at the geometry of the parabola. Recall that the height of any point of the parabola's surface is given by the expression r²/2R. And Calculus tells us the slope is given by r/R. The direction the reflection heads in depends on that slope.
Extend a line perpendicular from the surface, and find out where it intersects the centerline.
Since the slope of the line from A to D is r/R, and the distance from B to D is r, the distance from A to B is r/(r/R) = R. So the distance from A to C is R+r²/R. Now put a pinhole source at a distance R from the mirror, and shine the light at point D. The light will reflect back symmetrical about the line A-D. The light source is at E, a distance R from point C on the surface (because we put it there). The light shines to point D on the surface and reflects back to F. Since A to C = R+r²/2R, and the distance from E to C is R, the distance from A to E is r²/2R. The distance from A to F is only very slightly greater than r²/2R, so from E to F is approximately twice that, or r²/R.

What we will do is find the point F by introducing a knife edge from the side. If the knife edge hits F, then the image on the mirror at point D and D's twin on the left side of the mirror will both turn grey at the same time. Then we will use the measurement of the distance from E to F to determine the shape of the mirror.

More detail to come...

How to use the Foucault Test