F4 (E2, E6, E4)
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 2.17 Hierarchical 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 2.18 Network structure of a pyramid object.
2.3.2 Curvy Geometric Elements
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 2.3 Representation of 2D and 3D curves.
| Curvy features | Representation | Example | ||
| 2D curve | Explicit |
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| Implicit | f(x, y) = 0 | (x − x0)2 + (y − y0)2 = R2 | ||
| 3D curve | Explicit |
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| Implicit |
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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 2.3 and 2.4 can be readily expanded to represent 3D surfaces as follows:
Table 2.4 High‐order curves.
| Interpolation | Representation | Illustration |
| Lagrange |
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| Bezier |
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| Cubic spline |
P(t) = a1 + ta2 + t2a3 + t3a4 where t = [0, 1] and the coefficient vectors a1, a2, a3, and a4 are selected to satisfy |
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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 2.19 can be represented mathematically as
Figure 2.19 Representation of a spherical surface.
In an explicit form:
(2.14)
In an implicit form:
(2.15)
In computer aided geometric modelling, 2D and 3D
