Zhuming Bi - Computer Aided Design and Manufacturing

The information of solids must be stored in computers. Three popular methods to represent the data structure of solids are relational structure , hierarchical structure , and network structure .

A relational structure defines a set of lists for vertices, edges, and faces, and these lists are stored in term of arrays in a relational database. Taking an example of the pyramid in Figure 2.16, its rational structure includes the lists of vertices, edges, and faces shown in Table 2.2.

Figure 2.16Relational structure of a pyramid object.

Table 2.2Relational structure of pyramid object.

A solid can also be represented by a hierarchical structure where geometric elements are organized by a tree‐like layer structure. For a pyramid object, the highest layer is the solid body, which is bounded by four faces, each face being bounded by three edges and each edge being defined by two vertices. The complete hierarchical structure of a pyramid object is shown in Figure 2.17.

Figure 2.17Hierarchical structure of a pyramid object.

A hierarchical structure includes redundant information since the same geometric elements may have relations with multiple elements at high levels. Such redundancies can be eliminated by using a network data structure. As shown in Figure 2.18, a network data structure uses data pointers to represent the topological connections of geometric elements. The number of data pointers for a geometric element type can be varied based on the number of connections the element has with others.

Figure 2.18Network structure of a pyramid object.

Most objects have curvy boundary edges and surfaces. A curvy edge has one independent variable. As shown in Table 2.3, a curvy edge in 2D and 3D can be represented explicitly or implicitly in terms of a normalized length variable t from the starting point to the ending point.

Table 2.3Representation of 2D and 3D curves.

Curvy features	Representation	Example
2D curve	Explicit
Implicit	f ( x , y ) = 0	( x − x 0) 2+ ( y − y 0) 2= R 2
3D curve	Explicit
Implicit

The complexity of a 3D curve can be measured by the order of polynomial terms in its mathematic model for piecewise interpolation. Given a number of control points on the curve, different interpolation methods lead to different results for 3D curves.

The mathematic models for 2D or 3D curves in Tables 2.3and 2.4can be readily expanded to represent 3D surfaces as follows:

Table 2.4High‐order curves.

Interpolation	Representation	Illustration
Lagrange
Bezier
Cubic spline	P ( t ) = a 1+ ta 2+ t 2 a 3+ t 3 a 4where t = [0, 1] and the coefficient vectors a 1, a 2, a 3, and a 4are selected to satisfy

In an explicit form :

(2.12)

where u , v are normalized independent variables of surface.

In an implicit form :

(2.13)

An example of a spherical surface in Figure 2.19can be represented mathematically as

Figure 2.19Representation of a spherical surface.