The calculation of the solid angle subtended by a given surface is required in a wide variety of applications, ranging from optics to particles transport. Integration via a Monte Carlo process is prohibitive from the computational point of view even with state-of-the-art variance reduction techniques. Exact computation of the solid angle from the distance z between the emission and the detection plane and the lateral distance ρ to the detector center requires the two-dimensional integration of a density function. We develop an approximation strategy whose form optimizes the precision requirements and the computational speed. One way to achieve this simplification is to approximate the trigonometric function in the full description of the solid angle by a series of line segments. The approximation obtained gives results that are precise to at least 1 part in 1000 and that are as fast as known algorithms.
"Mathematical modeling of the solid angle function, part I: approximation in homogeneous medium," Optical Engineering 32(9), (1 September 1993). https://doi.org/10.1117/12.145056