The alignment telescope plus digital camera was deliberately slightly misaligned with respect to the collimator mechanical axis in such a way that the two wire reticles appeared clearly away from each other at the digital camera image, so a measurement of the separation between the two was possible. In this manner, the optical axis of the measurement system, $z\u2032$, is clearly inclined with respect to the collimator mechanical axis, $z$. The inclination angle between the two axes is given by the distance between the crossings of the two wire reticles, measured by the digital camera (on the $x\u2032y\u2032$ plane), divided by the distance between the crossings of the two wire reticles along the $z$ axis. Once we know the inclination angle, the distance from the wire reticles and the lenses, and the distance between the two lenses within the optomechanical mount (along the $z$ axis), we can write a coordinate transformation from those measured with the digital camera, in the inclined reference system, $S\u2032$ (on the $x\u2032y\u2032$ plane), to those in the designed reference system, $S$ (on the $xy$ plane). By means of a few simple geometrical calculations, it can be shown that the coordinates on the $xy$ plane may be expressed as a function of the coordinates measured on the $x\u2032y\u2032$ plane as Display Formula
$x=x\u2032\xb1\alpha (D+e),$(11)
Display Formula$x=x\u2032\xb1\alpha (D+e+d),$(12)
respectively, for the divergent lens (directly in front of the measurement system) and the convergent lens (behind the divergent lens), where $\alpha $ is the inclination angle between $z$ and $z\u2032$, $D$ is the distance from the closest wire reticle to the lenses to the first surface of the divergent lens, $e$ is the on-axis thickness of the divergent lens, and $d$ is the on-axis distance from the divergent and convergent lenses. Similar expressions can be found for the $y$ coordinate.