Solid modelling representations:

Mathematically specking, a set S is regular if and only if

                                                              S = k i S

Set operations (also known as Boolean operators) should be regularized to ensure that their outcomes are always regular sets. For geometric modelling, it means that solid models built up from well-defined primitives are always valid & represent valid (no-nonsense) objects. Regularized set operators preserve homogeneity & spatial dimensionality. The former refers that no dangling parts must result from utilizing these operators and the latter means that if two three-dimensional objects are combined by one of the operators, the resulting object must not be of lower dimension (two or one dimension). Regularization of set operators is specifically useful while users deal with tangent objects, as shall be seen shortly in an instance.

Depend on the above description, regularized set operators may be described as follows :

P ∪* Q = k_i (P ∪ Q)

P ∩* Q = k_i (P ∩ Q)

 P - *Q = k_i (P - Q)

c*P = k_i (cP)

Here the superscript * to the right of each operator indicate regularization. The sets P and Q utilized in Eqs. are supposed to be any arbitrary sets. Though, if two sets X and Y are r-sets (regular sets), that is always the case for geometric modelling, then Eqs. become

X ∪* Y = X ∪ Y

X ∩* Y = X ∩ Y ⇔ bX and bY do not overlap

X - *Y = k (X - Y)

c*X = k (cX)

Yet, the ability to edit models by adding, deleting ,replacing, and modifying subtrees coupled along with the comparatively compact form in which models are stored up, have made CSG one of the dominant solid modelling representations. The disadvantage of CSG is its inability to represent sculptured surfaces & half-spaces.