Solid Geometric Modelling

Half-Space Methods.

Suppose a surface divides the whole of three-dimensional space into two distinct regions. Each such region is called a "half-space". This does NOT imply that they are equal in volume. One possible surface is the sphere denoted by the equation

x*x + y*y + z*z = 1

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

x*x + y*y + z*z > 1

This is the entire volume outside the sphere. The other half-space is the small volume inside the sphere, indicated by the equation

x*x + y*y + z*z < 1

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

Set Operations

In order to define an object, we frequently have to combine several half-spaces. These may be done using the following set-operations.

Union of A and B

Union of A and B.

(denoted by AUB or BUA) This is the set of all points contained in either A or in B or in both of them. It is indicated as case (a) in the figure, where the heavy outline indicates the result and the dotted outlines indicate the objects A and B.

Intersection of A and B

Intersection of A and B.

This is the set of all points which lie in both A and B. If the two volumes do not overlap, then the intersection will be the empty set. This is shown in case (b) of the figure.

Difference A-B or B-A.

Difference A-B.

The difference A-B is the set of all points which lie in A but not in B. It is shown in case (c) of the figure, while case (d) shows the difference B-A.

Difference B- A.

These set-operations are used to combine the half-spaces to build up various objects. For example, let us consider the cube built up from the intersection of the following six half-spaces.

-4 < y and y < 4
-4 < x and x < 4
-1 < z and z < 1

This describes a rectangular cube, centred on the origin, whose boundary is the six planes whose equations are given below.

y = -4 and y = 4
x = -4 and x = 4
z = -1 and z = 1

Rectangular Cube.