Suppose I have two hemispheres in 3D space with centers and radii c1, r1 and c2, r2. The planes that define the bases of the hemispheres have coefficients (A1, B1, C1, D1) and (A2, B2, C2, D2). I am interested in calculating the intersecting volume if these two hemispheres clash.
I developed a approximate method to do this, which basically breaks down the two hemispheres into equally spaced points in a common coordinate system. Then, I estimate the volume by calculating the number of points that are present in both the hemispheres. The smaller I make the spacing, the more accurate the answer. But reducing the spacing increases the runtimes too. I am looking to use a analytical way to obtain faster and more accurate answers.
I know that for two intersecting spheres, the intersecting volume is given by the equation shown in http://mathworld.wolfram.com/Sphere-SphereIntersection.html and the volume of two intersecting hemispheres is some fraction of this. I have tried to use the angle between the base planes to provide some indication of the fraction but have failed to do so.
Sorry I am not a math major and may be missing out on some geometry basics.
Thanks