Regularized Set Operators:

CSG does not provide a unique representation. The set operations (⊂, ∪, ∩, and -) covered in the previous section are also known as the set-theoretic operations. While we use these operations in geometric modelling to make complex objects from primitive ones, the complement operation is generally dropped because it might make unacceptable geometry. Furthermore, if we utilizes the other operations (∪, ∩, -) without regularization in solid modelling, they can cause user inconvenience. Additionally objects resulting from these operations can lack geometric closure, can be difficult to validate, or can be inadequate for application.

To ignore the above difficulty, the point sets that represent objects and the set operations that operate on them should be regularized. Regular sets and regularized set operation (Boolean operations) are assumed as Boolean algebra.

A regular set is described as a set that is geometrically closed. The notion of regular set is introduced in geometric modelling to make sure the validity of objects they represent and thus eliminate nonsense objects. Under geometric closure, a regular set contain interior and boundary subsets. More significantly, the boundary contains the interior and any particular point on the boundary is in contact along with a point in the interior. In other terms, the boundary work as a skin wrapped around the interior.