Template Coordinate Systems


8/22/01
Information about coordinate systems for the template and Beam3.

Starting with the two given pieces of information in this analysis, the metal template used for aligning the SPARO/Viper system and the Beam4 ray tracing file, it is desireable to create some way of relating the two. This document details the attempts to do just that.

At the beginning of the analysis of the tertiary (detailed here), the following Beam4 ray trace file was used:

6 rays      sparo.ray
  Y0      X0     Z0   @wave
------:--------:----:-------:
 0.000:1.300000:-1.0:       B
 0.000:1.000000:-1.0:       :
 0.000:1.600000:-1.0:       :
-0.350:1.300000:-1.0:       :
-0.350:1.300000:-1.0:       :
Additional columns were used to glean output data from the ray trace, however these four columns were the only input data. This was the file used by Tom Renbarger to do the preliminary investigations into problems with SPARO's alignment. As one can see, the first ray, at Y=0, X=1.3 is the center ray. For all intents and purposes, the X value is the amount that the ray enters the system off the optical axis.

In my initial analysis, I naievely took the first ray in the ray table to be the center ray of the system. I also assumed that the prime focus of the primary will be a single point where the light rays converge after being reflected by the primary. Thus it would be possible to measure everything relative to a coordinate system with the prime focus at the origin and the CR as the y-axis. I called this the center ray coordinate system (CRCS). Using this setup however revealed serious problems with the definition of the center ray, as detailed in this report in point one. Moreover, the primary is aplanatic, yielding a 'blurry' prime focus that is corrected by the secondary. Thus, it is hard to define prime focus.

Thus we needed a different coordinate system if we were to make any headway.

Dr. Novak suggested that we choose the chopper as the x-axis, use the 'prime focus' on the template as the origin (it really is a single point, not a blurred out point) and measure things this way. This is the chopper coordinate system (CCS). Operating under the assumption that the template matches the real-life SPARO system and that the Beam4 representation is a mock-up of the real-life system, measuring points on the template, and then rotating them, should yield points that line up on the various surfaces in Beam4. This is true, to within a as yet undetermined translation (due to the uncertainty in the location of the 'prime focus'). The translation is only a few millimeters though, and is roughly comparable to measurement error in this experiement, and thus has little real effect.

The rotation is based off the rotation of the chopper in the Beam4 file, which is, according to the file used, 41 degrees from vertical. To get this to correspond to the Beam4 Z and X axis, a rotation of 131 degrees (41+90) clockwise of the axes is needed. Thus, given any point (z,x), the expression to move it from the CCS coordinates to the Beam4 coordinates is as follows:

 [z']     [ cos(131)  -sin(131) ]     [z]
([  ]) = ([                     ]) * ([ ])
 [x']     [ sin(131)  cos(131)  ]     [x]
See Howard Anton, Elementary Linear Algebra, 1984, 207-210 for an introduction on this. (Of course, most all algebra books have something on this subject.)

Using the method described above, the chopper and the secondary mirror both translate well up to a small translation from template to Beam3. The tertiary is a little more complicatied, but this can be for a number of reasons that are still under investigation.


Last updated August 22, 2001 by C. Greer.