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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	If a surface cannot be described by analytic or moving curves, they are called free‐form or sculpture surfaces. Control points are used to determine the surface. The mathematic presentation of these surfaces is similar to the spline curves. The parametric surface description uses two independent variables (u, v).

Geometric modelling is used to create a virtual geometric representation of a real or imagined object, which includes information of the shape, dimensions, and materials of an object. Many methods have been developed to model geometrics of products. A better understanding of the theoretical basics of geometric modelling helps in (i) improving modelling efficiency and (ii) shortening the learning curves of various CAD systems. While every method has its limitations, no universal solution is available that satisfies all demands for geometric models in itself. To select a modelling method, one must ensure the validity of the geometries.

A manifold is a topological space that locally resembles a Euclidean space near each point. In an n‐dimensional manifold, any neighbourhood of a point is homoeomorphic to the Euclidean space of dimension n. Accordingly, computer geometric models can be classified based on manifolds or non‐manifolds. Table 2.6 shows a few examples of manifolds or non‐manifolds; a non‐manifold usually includes some entities of different dimensions (1D, 2D, or 3D).

Table 2.6 Manifold and non‐manifold examples in 1D, 2D, and 3D.

N‐dimension	Manifold example	Non‐manifold example
1D (line)			The intersecting point is not locally homogeneous.
2D (surface)			The intersecting line is not locally homogeneous.
3D (solid)			The geometry mixes 2D and 3D entities.

In a computer representation, the information about physical objects is digitized. In other words, a free‐form curve or surface is represented by a set of straight‐line segments or flat patches. The geometric topology concerns the connectivity of geometric elements. As shown in Figure 2.20, not all geometric elements can be connected together for a valid geometric topology.

Figure 2.20 Examples of valid and invalid geometries. (a) Same geometries with different topologies. (b) Different geometries with the same topologies. (c) Invalid geometry.

A valid polyhedral in a 3D space should be homomorphic to a sphere and the validity of the geometry can be evaluated using the Euler–Poincare Law as

(2.16)

where F, E, V, B, L, and G are the numbers of faces, edges, vertices, bodies, inner loops on faces, and genuses in a geometry, respectively.

The meanings of inner loops and genuses are illustrated by the examples in Figure 2.21.

Figure 2.21 Inner loop and genus examples. (a) Inner loop example. (b) Genus examples.