Problems of radiation pattern synthesis, sidelobe suppression, or back radiation suppression appear when using antennas with reflectors or ground planes. The use of a semitransparent surface allows the required shape of a radiation pattern to be synthesized efficiently through the variation of reflection and transmission coefficients. Effective radiation pattern synthesis is possible when using analytical formulae designed for the radiation patterns of antennas with semitransparent reflectors. In this paper, we present a model for the scattering of a toroidal wave of a general form by means of a semitransparent reflector of revolution. The model allowed us to simulate the radiation patterns of various kinds of axisymmetric antennas with semitransparent reflectors and ground planes. Two asymptotic expansions of the radiation pattern of the model were found in the physical optics approximation. The expansions allow the radiation pattern for the entire space to be determined. As an example of its application, we used the model to calculate the radiation patterns of a plane with Archimedean two-wire spiral antenna and a loop antenna above concave and convex reflectors, as well as an Archimedean slot spiral antenna with a ground plane. We synthesized the optimal profiles of perfectly conducting and semitransparent reflectors and ground planes to improve the front-to-back ratio of the antennas.