Define the following variables:

**F**- The focal length (mm).
**$\psi $**- The sky position angle in radians.
**$\psi $**_{0}- The offset between the coordinate system of the guide probe and that of the instrument rotator, in radians.
**$\epsilon $**- The handedness of the guide-probe coordinate system ($\pm $1).
**x y**- Guide probe coordinates, in encoder units.
**x$$**_{min}x$$_{max}y$$_{min}y$$_{max}- xy limits on the area within which guide stars may be selected, in encoder units.
**r $\theta $**- Polar coordinates of the guide probe (encoder units; r is positive by definition).
**$\theta $**_{min}θ_{max}r$$_{min}r$$_{max}- Limits on the area within which guide stars may be selected, in polar coordinates; encoder units).
**e$$**_{r}e$$_{θ}- The linear and angular encoder scales in mm/encoder unit and rad/encoder unit, respectively.
**x$$**_{0}y$$_{0}- The position of the centre of rotation in encoder units in the guide probe coordinate system (used for xy systems only).
**r$$**_{0}- The radial offset for guide probe coordinates, in encoder
units (used for r$\theta $ systems only).

All of the existing A&G boxes on the La Palma telescopes use xy stages for their guide probes, with the exception of the WHT Cassegrain system, which has an r$\theta $ arrangement. A general description of the search areas may be given as follows (note that not all configurations use every constraint):

- $x$
_{min}< x < x_{max}and $y$_{min}< y < y_{max}(within a rectangle). - $[(x-x$
_{0})^{2}+ (y-y_{0})^{2}]^{1/2}≤r_{max}or $r\; \le r$_{max}(within a circle: this may be either a result of vignetting or a restriction on the mechanical travel of the probe). - $(x-x$
_{0})^{2}/l^{2}+ (y-y_{0})^{2}/m^{2}> 1 (inner ellipse excluded). A minimum radial travel for an r$\theta $ system is a special case. A similar condition, $[(x-x$_{0})^{2}+ (y-y_{0})^{2}]^{1/2}> r_{min}or $r\; >\; r$_{min}is used for the`ROTATOR ANY`option, where the available area is always a circular annulus. - $x\; <\; x$
_{}- or $x\; >\; x$_{}+ or $y\; <\; y$_{}- or $y\; >\; y$_{}+ (*i.e.*the area $x$_{}- < x < x_{}+ and $y$_{}- < y < y_{}+ is*excluded*). - The probe azimuth is in the range $\theta $
_{min}< θ< θ_{max}.

Mon Mar 1 17:52:09 GMT 1999