Let's say I have two spheres whose center's coordinates (cartesian) are


and both have radius R.

I want to analytically calculate the total surface area of both spheres. When d>2R it is obviously two times the surface of one sphere, so 8*Pi*R^2. When d is smaller than 2R, spheres overlap, but using surface of revolution integrals, I can easily derive the expression for the surface area.

Before I had a "one-dimensional" array of spheres. But in a "two-dimensional case", let's say I have four spheres, with coordinates


when d is smaller than 2R (and therefore overlapping spheres) I do not see clearly how can I calculate the total surface. And same problem in the case of 8 spheres in a similar "three-dimensional" arrange.

Anybody could give me any hints?


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