The optimal design of a radiotherapy treatment depends on the collection of directions from which radiation is focused on the patient. These directions are manually selected and are based on the treatment planner’s experience. Once the angles are chosen, there are numerous optimization models that decide a fluency pattern (exposure times) that best treats a patient. So, while optimization techniques are often used to decide how long a patient will be exposed to a high-energy particle beam, the directions themselves are not optimized. The problem with optimally selecting directions is that the underlying mixed integer models are well beyond our current solution capability. We present a rigorous mathematical development of the beam selection problem that provides a unified framework for the problem of selecting beam directions. This presentation provides insights into the techniques suggested in the literature and highlights the difficulty of the problem. We also compare several techniques head-to-head on two-dimensional problems.