We certainly expect that the union, intersection and difference
of two solids is a solid. Unfortunately, in many cases this is
not always true. In the following figure, two cubes touch each
other and their intersection is a rectangle shown on the right.
A rectangle is not a three-dimensional object and hence not a
solid! To eliminate these lower dimensional branches, the three
set operations are regularized as follows. The idea of
regularing is very simple. Let +, ^ and - be the regularized set
union, intersection and difference operators. Let A and B be two
solids. Then, A + B, A ^ B and A - B can be defined
mathematically based on the above description: A + B =
closure(int(the set union of A and B)where closure() and int()
are the closure and interior discussed in previous page. Based
on this definition, the intersection of the two cubes shown at
the beginning of this page is empty. As mentioned earlier, the
set intersection of these two cubes is a rectangle, which is a
two dimensional object and has no int