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.

- Compute the result as usual and lower dimensional components many be generated.
- Compute the interior of the result. This step removes all lower dimensional components. An example has been shown in previous page. The result is a solid without its boundary.
- Compute the closure of the result obtained in the above step. This adds the boundary back.

Let **+**, **^** and **-** be the *regularized* set union,
intersection and difference operators. Let ** A** and

where=A+Bclosure(int(the set union ofandAB)

=A^Bclosure(int(the set intersection ofandAB)

=A-Bclosure(int(the set difference ofandAB)