Geometric differencing
In the realm of constructive solid geometry, questions like the one you asked can probably be answered. Here, solids are constructed by unions and differences of basic shapes (for example, a sphere with a tunnel can be described as Ball - Cylinder. Shapes are then built up as trees of operations on basic shapes (at the leaves), and the differencing problem can then be thought of as a labelled tree-diff problem (which has been examined among other things in the context of comparing XML documents).

I'm not claiming that these questions are already being studied, but CSG is definitely one place where the geometric questions you're asking might be addressed.