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 |
|
|
|
| Implicit | f(x, y) = 0 | (x − x0)2 + (y − y0)2 = R2 | ||
| 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 2.3 and 2.4 can be readily expanded to represent 3D surfaces as follows:
Table 2.4 High‐order curves.
| Interpolation | Representation | Illustration |
| Lagrange |
|
|
| Bezier |
|
|
| 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 |
|
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 curves are commonly used to generate 3D surfaces. Depending on surface features, 3D surfaces can be classified into swept, ruled‐generated, or free‐formed surfaces, as shown in Table 2.5.
Table 2.5 Types and features of surfaces.
| Surface type | Feature | Illustration |
| Swept | A swept surface is defined by two elements, i.e. driving curve (D) and guide curve (G) or trajectory. The driving curve (D) can be open or closed. The guide curve (G) will run along D with the constant contact point. |
|
| Ruled | A ruled surface is defined by three 3D curves. The G curve drives along the D1 curve and leans in D2. In the first case, D1 and D2 are divided into equal segments and the end points of these segments are connected by G. In the second case, the G curve just leans to D1 and G will be parallel in every position. Other variations can be generated from a ruled surface by application of a non‐constant G curve. |
|
|
Freeform generated
|