I have two spherical shells (green and red), given by the following radii and the centers.
green inner: g_i
green outer: g_o
green center: (gc_x, gc_y, gc_z)
red inner: r_i
red outer: r_o
red center: (rc_x, rc_y, rc_z)
The first illustration shows the problem statement from the side view, in the second in 3D, where the area I am interested in, is green filled.
Source: The diluted aqueous solvation of carbohydrates as inferred from molecular dynamics simulations and NMR spectroscopy, Fig.6
Background information: I work with two robots, where the interaction range is defined by the mentioned radii for each robot individually. The calculation is for checking whether their is an intersection of their interaction ranges, which are defined by the area (side plot) in range (g_i, g_o) respectively (r_i, r_o) respectively by the green area (3D plot).
Question
I would like to get the mathematical representation of the intersected 3D volume area (described with the orange arrow), where the areas [g_i, g_o] overlaps with [r_i, r_o], which I can use for the mentioned application.