An extension of Taylor's circular aperture pattern synthesis technique permits the generation of a flat-topped beam that is rotationally symmetric, surrounded by ring side lobes of controllable height, with the ripple amplitude also controllable. The corresponding continuous aperture distribution can be made pure real by appropriate root-pairing, which makes the results applicable, via sampling, to resonantly-spaced waveguide slot arrays. A one-dimensional linear stretch extends the results to rectangular grid planar arrays possessing an elliptical boundary. In that case the pattern is a fiat-topped beam with an elliptical contour, surrounded by ring side lobes also of elliptical contour, but with amplitude and side lobe level still controlled. The principal application is to space vehicle communication systems required to place a footprint pattern on a selected portion of the earth's surface.