Читать книгу Computer Aided Design and Manufacturing - Zhuming Bi - Страница 64

Boundary surface modelling (B‐Rep) is used to define the finite and closed cover of an object (the mantle) upon a surface model. In B‐Rep modelling, it is assumed that each physical object has an unambiguously determinable boundary surface, which is a continuous closing set of surface patches. Since a finite volume is defined, the B‐Rep method provides a comprehensive topological characterization of solids.

In defining the volume of solids, B‐Rep modelling utilizes a surface model with all of its determined boundary surface patches, and the normal vector of each surface patch is then determined. A watertight volume can be finally determined by collecting all spatial points at the internal sides of all the surrounding boundary surfaces. This is implemented by the half‐space concept.

Mathematically, 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 H_i (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.