a half space can be expressed as
(2.17)
which shows that point P belongs to a half space E3 if the condition f(P) < 0 can be satisfied where f(P) = 0 is the equation of the surface in an implicit form. The examples of surface equations for some common geometric elements are given in Table 2.10.
Table 2.10 Equations of common surfaces.
| Name | Illustration | Implicit equation f(P) = 0 |
| Flat XY plane |
|
{(x, y, z) : z = d} |
| Cylinder |
|
{(x, y, z) : x2 + y2 = R2} |
| Cone |
|
{(x, y, z) : x2 + y2 = kz2} |
| Sphere |
|
{(x, y, z) : x2 + y2 + z2 = R2} |
| Torus |
|
{(x, y, z) : (x2 + y2 + z2 − R22 − R21)2 = 4R22(R21 − z2)} |
The half space on one side of the surface is empty while the other one is filled with a material. The B‐Rep method assumes that the volume occupied by an object is bounded by surfaces of infinite extension. Each surface divides space into two regions of infinite extension. The volume of the solid S is the intersection (common portion) of half spaces Hi (i = 1, 2, …, N) where
(2.18)
Figure 2.25 shows an example where the B‐Rep method is applied to define a rectangle body (Figure 2.25b) as an intersected volume of seven surfaces of infinite extension (left, right, rear, front, top, bottom, and central cylinder) (Figure 2.25a).
Figure 2.25 Example of half‐spaces in B‐Rep method. (a) Seven surfaces with an infinite extension. (b) Enclosed volume by seven half spaces.
2.4.4 Space Decomposition
In a space decomposition, an object is represented by a collection of isomorphic cells. The sizes of isomorphic cells are very small, being several orders of the magnitude smaller than the dimensions of an object. The space decomposition method is very popular in numerical simulations such as finite element analysis and boundary element analysis.
Figure 2.26 illustrate how the space decomposition is processed as well as the corresponding data structure. The space decomposition is performed as the following procedure:
1 It divides a finite space into eight parts (producing octants).
2 It then examines each space region as to whether they are fully or partly filled.
3 Partial regions that are totally filled up or are not filled at all can be excluded from further investigations.
4 The partially filled octants are continuously refined until the required accuracy is achieved.
Figure 2.26 Data structure of space composition.
Figure 2.27 shows examples of using the space decomposition method to represent objects (Bi and Kang 2014). In the left column, the datasets of point clouds are obtained by 3D scanning, and in the right column, the point clouds have been transformed into solid geometries using the space decomposition method.
Figure 2.27 Examples of solid objects using a space decomposition method (Bi and Kang 2014). (a) Point cloud datasets of objects. (b) Solids and surfaces from the space decomposition method.
In the numerical simulations, the finite volume of solid is decomposed into small cells, so‐called isomorphic cells; these cells are usually smaller by several orders of magnitude than the dimensions of the solid itself. The space decomposition in a numerical simulation is also called a meshing process and Figure 2.28 shows two example models where the solids are decomposed into a set of small cells called elements.
Figure 2.28 Examples of space decomposition in numerical simulation. (a) Solid A. (b) Solid B.
2.4.5 Solid Modelling
In solid modelling, the geometry of an object is modelled by a set of solid primitives, which are assembled into an object using composition operations. The modelling procedure to combine elemental solids using composition operations is commonly known as Constructive Solid Geometry (CSG) modelling. The geometric representation from solid modelling gives complete information about physical objects. CSG modelling is based on the following assumptions:
1 A constitutive object is a rigid solid; the object has a concrete and invariant shape not affected by spatial location or position.
2 An object fills the space occupied by it homogeneously; the inside positions
