The "surface" exists only in your mind. All you have are points. Isolated points. So until you find away to connect those points into a true surface in some form, you can do nothing.
Simple is to use a convex hull. That is great, if the volume described is truly convex, or even sufficiently close to being convex. Your next choice is to use an alpha shape.
In either of the above cases, the outer boundary is a triangulation. So now the problem reduces to solving for the intersection of a line with a triangulated surface. This part is pretty easy. You just need to solve for the intersection of a line with a triangle, and then repeat for each triangle. Better code will see how to eliminate some of those triangles from consideration, but in any case it is not that hard. Just computational geometry.