Using projection operators, we rederive x-y geometry discrete ordinates-to-spherical harmonics (SN → PN-1.) fictitious sources defined in the literature as ray-effect mitigating devices. We define a new x-y geometry fictitious source with certain properties that are superior to earlier sources. A detailed description of the S2 → P1 source, including a discussion of vacuum and reflective boundary conditions, is provided. We then derive fictitious sources in r-z geometry that give spherical harmonics and spherical-harmonics-like solutions. Finally, a simple algorithm is presented that allows a significant reduction in the iteration time needed to obtain ray-effect-free solutions. This algorithm effectively reduces the size of the fictitious source in energy groups where ray-effect distortions are not expected. The new sources and the algorithm for reduction of computation time make this approach viable for solving the ray-effect problem.