HOW TO SET UP AN AUXILIARY OPTICAL AXIS
                      (c) 1992, 2003, 2009 M.Lampton
                           STELLAR SOFTWARE
 

     Suppose a group of optical elements is to be lined up along
     some axis that is neither the lab X, Y, or Z directions.  How
     does one express the positions and orientations of its elements
     in the lab frame to enter them into a BEAM optics table?

     This auxiliary axis will have its origin somewhere in space,
     say (Xa Ya Za).  It will also have some direction along which
     the optical elements are aligned.  There are three ways to set
     up the auxiliary axis, depending on what you initially know
     about its direction:

        A: the angles (tilt, pitch) of its normal are known. Tilt is
        the rotation of a surface about the lab X axis; pitch is the
        rotation about the tilted Y axis.

        B: its direction cosines (U, V, W) are known.  These are
        the projections of a unit vector along the auxiliary axis
        onto the lab X,Y,Z directions.

        C: a second point (Xf Yf Zf) along the axis is known.


     CASE A:  The pitch and tilt numbers are used by BEAM to
     orient the optical components so that they are normal to the
     auxiliary axis.  Put these into your optics table in your pitch
     and tilt columns.  Compute U,V,W using the formulas

                U = sin(pitch)
                V = -sin(tilt) * cos(pitch)
                W = cos(tilt) * cos(pitch)

     and then position each element using its known distance D from the
     origin of the auxiliary axis (a spreadsheet is useful here):

                (X0+U*D1, Y0+V*D1, Z0+W*D1)   for the first element
                (X0+U*D2, Y0+V*D2, Z0+W*D2)   for the second element
                (X0+U*D3, Y0+V*D3, Z0+W*D3)   for the third element
                (X0+U*D4, Y0+V*D4, Z0+W*D4)   etc.


     CASE B:  First find the pitch and tilt of all the auxiliary
     elements, using the formulas

                pitch = arcsin(U)
                tilt = arctan(-V/W).

     Put these values, in degrees, into your optics table under
     pitch and tilt column headers.  Copy these numbers down the
     page so that they apply to all the elements that are aligned on
     your auxiliary axis.  Then position each element along the
     auxiliary axis using the position formulas in case A.


     CASE C:  First find the direction cosines of your auxiliary
     axis using the Pythagorean theorem:

                D = sqrt((Xf-Xa)^2 + (Yf-Ya)^2 + (Zf-Za)^2)
                U = (Xf-Xa)/D
                V = (Yf-Ya)/D
                W = (Zf-Za)/D.

     Then, with these numbers known, go to case B and complete the
     pitch, tilt, and element positioning.