I have a lot of spherical coordinates (with radius = 1) and want to find out, if the angle between all coordinates and a reference coordinate is bigger or smaller then a reference angle. I do not necessarily need the actual angle. I want to do this in a preferably fast way as the calculation is repetitive and I try to display the results in "real time".
My first approach was to convert all spherical coordinates to cartesian coordinates and compare the dot products of the vectors. With spherical coordinates I only find ways to use trigonometric functions, which are expensive to compute. So I'm wondering is there any way to avoid trigonometry and still directly use the spherical coordinates for comparing angles?