In shaped offset dual-reflector antennas, special care must be taken to prevent crosspolarisation for linear polarisation and beam squint for circular polarisation. This problem is addressed by deriving the laws that determine the mapping of the feed pattern onto the aperture of the reflector. There is a presentation of the conditions that must be satisfied for a GO mapping to be ideal without crosspolarisation and beam squint. The paper even present equations which allow the crosspolarisation and phase errors in the reflector aperture to be calculated directly from the mapping function. As an example, this theory is used to specify and characterise offset dual-reflector antennas with both circular and elliptical apertures. The equation for the crosspolarisation is generally valid only for dual-reflector antennas, but can in many cases also be applied to multireflector systems to obtain useful approximate results. This is shown by the application to a trireflector system; the proposed dual-reflector feed of the spherical reflector of the Arecibo Observatory.