Conformal mapping and complex coordinates in Cassegrainian and Gregorian reflector antennas

In a Gregorian or Cassegrainian reflector antenna, the complex coordinate u-prime of an output ray is related to the corresponding input coordinate u by a bilinear transformation, u-prime = (au + b)/(cu + d). The properties of this transformation are discussed, its coefficients a, b, c, d, are derived, and the conditions that must be satisfied in order that symmetry be preserved are given. The conditions are expressed directly in terms of the parameters that specify the path of the principal ray, which is the ray corresponding to the feed axis. The results are directly related to well-known properties of stereographic projections, and they are shown to be useful in the design of multireflector antennas which minimize aberrations and cross-polarization.