I'd like to use an abstract algebra engine of some sort to define 3d
shapes and their intersections.

For example, I would like to create a structure for a Sphere that is
represented by (P - P0) dot (P - P0) <= R0*R0
where R0 and P0 are constants.

And the intersections of two Spheres, is represented by a system of
inequalities:

[(P - P0) dot (P - P0) <= R0*R0,
(P - P1) dot (P - P1) <= R1*R1]

I'd also like to be able to determine the intersection points of the
surface of this system with a Ray.

The actual goal is to create a Beam Casting project in Java.  Beam
Casting is like Ray Casting, but you start with a pyramid instead of a ray.

I've found a library called Java Algebra System 2.0, but I'm confused on
how to apply it toward my goal.  If there is anything more specifically
tailored to my problem, or someone can help me understand JAS, I'd
greatly appreciate the info.

Thanks,
Daniel.