Half-space Method:

Solid representation is depends fundamentally on the notion that a physical object divides an n-dimensional space Eⁿ into two regions: exterior and interior, separated by object boundaries. In terms of above notion, a solid model of an object is described mathematically as a point set S in 3-D Euclidean space (E³).

Suppose a surface divides the entire of three-dimensional space in two distinct regions. Each such region is known a "half-space". It does NOT imply that they are equal in volume. One possible surface is the sphere indicated by the equation

                                                X * X + Y * Y + Z * Z = 1

The surface indicted the boundary of the two half-spaces, one of which is the infinite region

            X * X + Y * Y + Z * Z > 1

It is the overall volume outside the sphere. The other half-space is the small volume inside the sphere, denoted by the equation

         X * X + Y * Y + Z * Z < 1

It is still a valid half-space, although it is so much smaller than the other.