In the design of illumination lenses, there is a fundamental incompatibility between the spherical geometry of light radiating outwards and the rectangular geometry of typical illumination targets, analogous to trying to fit a round peg in a square hole. This amounts to establishing a rectangular grid on the sphere, the perennial problem of map-makers. Here we apply a new pseudo-rectangular spherical grid, originally developed for parallel-processor simulations of semiconductor devices, to establish correspondence between source-grid cells and the rectangular cells of a target grid. This correspondence establishes a grid of deflections, whereby source rays are deflected so as to impact the proper cell on the target grid. For a given lens refractive index, each deflection is implemented by the angles of inclination the ray encounters going into and out of the lens, resulting in two grids of surface gradient values, for the inside and outside lens-surfaces. Central spines are obtained for these surfaces by a linear integration, after which adjacent rows are successively obtained in a lawnmower fashion, so as to heal any imcompatible cross-derivatives. Example lenses are illustrated.© (2006) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.