What is the volume of an arbitrary intersection of a cube and a cylinder?

Question

I need to find the volume of an arbitrary intersection of a cube and a cylinder. the sides of the cube ($a$) will always be less than the diameter of the cylinder, such that a cube can fit fully inside the cylinder. Anyone have any idea how to find the volume of the intersection with the cube at an arbitrary position?

I found this case which seems to be the right shape for the intersection but the solution stipulates that $a>r$ which will never be true in my case.

It is also possible to assume that the intersection is close to a wedge (ie $a<<r$) but this will not always be the case.

Note: You can assume the cylinder is infinite.

Answer 1

Citing from an answer to the sub case area of intersection of circle and square:

The formula will be a mess, no matter how you look at it. There will be a lot of cases, period.

I would strongly advise to apply numerical methods here. Or maybe calculating some points in your configuration space (which might be the set of all $(a/r, x, y, \alpha, \beta))$ and then resort to approximation.