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The problem of transforming a given primary feed pattern into a desired aperture field distribution through two reflections by an offset dual reflector system is investigated using the concepts of geometrical optics. A numerically rigorous solution for the reflector surfaces is developed which realizes an exact aperture phase distribution and an aperture amplitude distribution that is accurate to within an arbitrarily small numerical tolerance. However, this procedure does not always yield a smooth solution, i.e., the reflector surfaces thus realized may not be continuous or their slopes may vary too rapidly. In the event of nonexistence of a numerically rigorous smooth solution, an approximate solution that enforces the smoothness of the reflector surfaces can be obtained. In the approximate solution, only the requirement for the aperture amplitude distribution is relaxed, and the condition on the aperture phase distribution is continued to be satisfied exactly.