ATM Generalized Caustic

[Anthony Stillman <atmer@flash.net>]

For some time, I've wanted to derive a formula for the caustic of an
arbitrary function.  Several weeks ago I set about doing this.  After
proving unequivocally that pi equals 2 and in turn that 2 equals 0,  I
started to make progress.  That progress is reported here.

What is the caustic?

It helps me to imagine a big fast asphere.  Stuck to a point on the surface
is one end of a string.  Holding the string taught, I arrange it, such that
it is perpendicular to the mirror's surface.  Where the string is stuck,
the surface has an ROC.  Measuring from the mirror, along the string, I
mark a point that is "that" ROC away.  The set of marks made in this manner
from all the points on the mirror, is the caustic.

To use this to measure a mirror, the light source must be coincident with
the knife edge.  Not all caustic testers, measure the caustic.  The light
source may be stationary, or it may move with the blade, but be displaced
slightly.  Schroader (ATM III) introduces a factor of two to approximate a
stationary source.

Fair warning, math follows.

The surface of a parabolized primary climbs from center to edge.  That is,
the mirror has a sagitta.  A measure of how steep the climb is, is called
the slope, or first derivative.   And too, from center to edge, the climb
(the slope) gets steeper.  A measure of how much the steepness steepens, is
called the second derivative.

To find an equation for the caustic, I used a cross-section of my imagined
example above.  Adding a second string distal to the first (farther from
the mirror's center), I arrange it in the same manner as the first.  If the
two strings are stuck down to points infinitesimally close to one another,
then it is reasonable to hypothesize that they share the same point on the
caustic.  That is, the two strings (lines) intersect at the caustic.
Using this model, I solved for the point of intersection of the two lines,
i.e. a point on the caustic.

For an arbitrary function (x,f(x)) the caustic of that function is (xc(x),
yc(x)), where:

  xc(x) = x - (df/dx) * ((df/dx)^2 + 1) /  (d(df/dx)/dx)

  yc(x) = f(x) + ( (df/dx)^2  + 1) / (d(df/dx)/dx)

Further, the distance between the points (x,f(x)) and (xc(x),yc(x)) is
R(x), where:

  R(x) = ( (df/dx)^2  + 1)^(3/2) ) / (d(df/dx)/dx)

In above equations:
The first derivative of f(x) with respect to x is written:   df/dx
The second derivative is written:                          d(df/dx)/dx


The formula for a parabola with osculating radius R is

   f(x) = (1/2R) * x^2

When this is substituted into the generalized form, the equations for the
caustic reduces to:

   xc(x) = - x^3 / (R^2)

   yc(x) =  ( (3/2R) * x^3 ) + R

   R(x)  =  ( (x^2 +R^2)^(3/2) ) / (R^2)

Schroader reports a slightly different version of the equations for xc(x)
and yc(x).  To  model a fixed source he introduced a factor of two.  Also,
his value "X" is an additional two times my value xc(x).  This, because his
value is a  measure of the distance between a point on the caustic and the
reflection of that point on the opposite side of the optical axis.
Schroader's equation for "Y" does not include the added term of R present
in my equation for yc(x).  In effect, moving the caustic to the origin.
Finally, Schroader uses the letter "r" where I use the letter "x".

There is much more to this than I have written here.  For one thing,
calculating the caustic from the mirror isn't exactly what one wants to do.
Calculating the mirror from the caustic, that would be better.
Unfortunately, the generalized equation for the caustic doesn't lend itself
to an analytically solution.  There are other means.

Of course none of this is really relevant to the average ATM (as if there
is such a beast).  With the right software the meticulous need not be math
able, to rival in the arena of high precision.


Anthony