That is, why are you advocating referencing the focal point for UO (using A^2/4H) and the apex for BO (using D(D-A)/4S)?
That's not quite what I was saying. Because it is based on the equality of apparent angles, as a function of diagonal size and eye separation, A^2/4H can be applied with either BO or UO, and will give correct shift value - for all practical purposes - for any eye-to-diagonal separation H (disregard for the moment that H has been defined as focus-to-diagonal separation; I sort of run out of meaningful superscripts, and can't use subscripts in this format). On the other hand, (D-A)A/4S is based on the geometry with the eye at the apex, and can't be applied to any eye-to diagonal separation (it will still work for both BO and UO with eye at apex).
Again, D(D-A)/4S formula derivation involves redefining (A) which makes the formula an approximation (a very good approximation) but â€œmathematicallyâ€ the formula is flawed â€“ which was my point.
And you were correct. There is no practical consequences, but it is good to keep the concept clear. So I went through it again and got out with the following:
- the coordinate system solution, and the resulting shift value given by A/4F' - which is the basis for slighly rounded off shift relation (D-A)A/4S - is applicable when the starting point is the desired field of full illumination, which is the basis of determining apex-to-focus distance, focal ratio F' of the cone converging to the apex, and needed minor axis A to cover the field.
- when starting with a known diagonal, the above approach still can be used, but can't avoid slight inaccuracy due to the needed offset being defined in terms of F', and thus of field diameter and eye-to-focus distance, neither of which is accurately defined without knowing the offset value first. Better concept for this scenario is the simplest one, based on similar triangles, which defines BO offset as:
where "z" is the mirror sagitta z=D/16F (there is no appreciable change if the sagitta is omitted). With the BO offset known, the exact value of F' is given by:
The details are on my Newtonian diagonal page
- for any given eye-to-diagonal distance H", the exact offset for either BO or UO is given by [sq.rt.(H"^2 + A^2)-H"]/2; obviously, that requires known diagonal size. If the change in the eye-to-diagonal distance due to diagonal shift is neglected, this simplifies to a very close approximation - accurate for all practical purposes - (A^2)/4H".
These two relations are probably having most of practical importance, not only because they can be applied to collimation from the focal point level as well, but also because collimation from the apex is actually done from a point somewhat closer to the diagonal (so that the primary's reflection is visible inside the diagonal). While the shift difference due to the latter may be only of academic interest, it is good to have a clear concept.