I'd also like to see (more work) a ML envelope display with the ability both to change the conic and the ROC. I use FigXP for this and almost never the surface plot as the information ML produces tells you exactly how to correct the mirror based on zone-zone variances.

I think I sent Carl a copy of a spreadsheet I made back around 2002 with the ML mathematics in it (also Foucault based on Holleran's area centers). The ML math is pretty simple. This is the write up on it that I put in the spreadsheet, with the S&T reference:

A Graphical Approach to the Foucault Test

Millies-Lacroix

Sky and Telescope, February 1976, pgs 127-129

In Lunettes et Telescopes, Paris, 1935, A. Danjon and A. Couder recommend that a mirror be considered finished if it satisfies this double criterion:

1) The geometric image of least aberation in the plane of focus should not exceed the size of the theoretical Airy disk.

2) The maximum wavefront error must not exceed a quarter of a wavelength of light, and the defects should be much less than this over most of the surface.

On page 141 in How to Make a Telescope, Jean Texereau gives the residual longitudinal error at the center of curvature as:

L_{c} = m - c - r2/R where m is the measured value, c is a constant subtracted from the measured values for all zones, r is the mean radius of the zone, and R is the radius of curvature of the mirror.

He also states that the longitudinal error at focus, L_{f}, is one fourth the above error, or: L_{f} = L_{c}/4, and shows that from the geometry the transverse aberration is to a good approximation:

l_{f} = L_{f} * r/f = (r/4f)(m - c - r2/R) from the above equations.

Millies-Lacroix imposes the first above condition, l_{f} < q, on this equation, where q is the radius of the Airy disk given by q = 1.22*l*f/D, f is the focal length, l is the wavelength of light (taken as 5600 A), and D is the optical diameter of the mirror. This gives:

r/2R(|m - c - r2/R|) < q, where |m - c - r2/R| denotes the absolute value of the difference as some values will be negative, some positive, depending on the slope of a particular zone relative to that for a perfect parabola.

Solving for m - c - r2/R:

m - c -r2/R < + 2Rq/r for m - c - r2/R > 0, and m - c - r2/R > - 2Rq/r for m - c - r2/R < 0

Millies-Lacroix then plots the values m - c - r2/R and +/- 2Rq/r. The values m - c - r2/R must fall within the envelope given by +/- 2Rq/r to satisfy the above inequality. If the mirror is a perfect parabola the aberation is r2/R, so m - c - r2/R will be zero for all measured values, and thus plot as a straight line on the x axis.

It's easy to create the plot in a spreadsheet, but a lot more work I expect for Carl to add it to his code.

**Edited by tommm, 11 May 2020 - 10:51 AM.**