Assume I have a world coordinate frame $\mathbf{w}$.
Assume I have a second coordinate frame that can be parameterized as a $4\times4$ homogenous transformation matrix with respect to the world coordinate frame: $\mathbf{^{w}T_{s_1}}$.
The origin of the first sphere $s_1$ has a 6DoF pose represented by the transformation matrix $\mathbf{^{w}T_{s_1}}$.
$s_1$ can be parameterized in spherical coordinates (mathematical convention) as $(\rho_1, \theta_1, \phi_1)$ where $(0 \leq \theta_1 \leq 2\pi)$ and $(-\frac{\pi}{4} \leq \phi_1 \leq \frac{\pi}{4})$. The restriction of $\phi_1$ means that $s_1$ is in fact a partial sphere that looks like a round bottomed cone. The central axis of the cone is pointing in the direction of the Z-axis of coordinate frame $\mathbf{^{w}T_{s_1}}$.
Assume I have a third coordinate frame that can be parameterized as a $4x4$ homogenous transformation matrix also with respect to the world coordinate frame: $\mathbf{^{w}T_{s_2}}$.
The origin of the second sphere $s_2$ has a 6DoF pose represented by the transformation matrix $\mathbf{^{w}T_{s_2}}$.
$s_2$ can be parameterized in spherical coordinates (mathematical convention) as $(\rho_2, \theta_2, \phi_2)$ where $(0 \leq \theta_2 \leq 2\pi)$ and $(-\frac{\pi}{4} \leq \phi_2 \leq \frac{\pi}{4})$. The restriction of $\phi_2$ means that $s_2$ is also a partial sphere that looks like a round bottomed cone. The central axis of the cone is pointing in the direction of the Z-axis of coordinate frame $\mathbf{^{w}T_{s_2}}$.
- What is the volume of intersection of the partial spheres $s_1$ and $s_2$?
It seems that given the coordinate transformations and different axis orientations, I may need to convert the spherical coordinates back to cartesian coordinates and then try to solve but I am not sure.
Thanks!