A method is presented for placing line, point, and ring light sources to produce uniform illumination of planar surfaces. Optimization consists in setting to zero as many terms as possible in a two-dimensional Taylor series expansion of the surface illumination. We analyze several practical lighting arrangements and discuss the significance of symmetry in lighting design. Four conditions are sufficient to produce optimal illumination: (1) place all lamps in a single plane parallel to the illuminated surface; (2) arrange linear sources in parallel pairs; (3) arrange point-source lamps to achieve fourfold symmetry with respect to any two orthogonal axes lying in the surface, this requires a minimum of four lamps; and (4) select lamp heights so that a line connecting each lamp to the center of the illuminated surface forms specific angles with the surface normal: 30 deg for line sources and approximately 39 deg for point and ring sources. Any lamp arrangement meeting these conditions produces an illumination function in which at least three orders of terms of the Taylor series expansion are zero. We also discuss circumstances under which some of these conditions can be relaxed.