Efficient computation of the solid angle function and its derivatives

The solid angle subtended by the photocathode with respect to a scintillation location can be seen as the most likely (normalized) response of a given photomultiplier. Integration via a Monte Carlo process is prohibitive from the computational point of view. Exact computation of the solid angle from the distance to the PM plane and the distance to the PM center requires the two-dimensional integration of a density function. The authors choose to find an approximating function whose form would permit further development of the model while providing a dramatic computational advantage. One way to achieve this simplification is to approximate the trigonometric function in the full description of light distribution by a series of line segments. The approximation obtained gives results that are precise to at least 1 part in 1000. Evaluation of the derivative follows the same general approach