机械CADChapter 3 Surfaces

3.1 Introduction
The wireframe models introduced in the previous chapters are not enough to represent the complex objects. The following example shows that the wireframe model in Fig. can be interpreted by three different ways, as shown in Fig. 3.1b, c and d.
∂P ∂x ∂y ∂z Pu ( u , v) = = i+ j+ k , u min ≤ u ≤ u max , v min ≤ v ≤ v max ∂u ∂u ∂u ∂u
(3.2)
And
∂P ∂x ∂y ∂z Pv (u , v) = i+ j+ = k , u min ≤ u ≤ u max , v min ≤ v ≤ v max ∂v ∂v ∂v ∂v
(3.3)
The two equations can be written as
∂x Pu ∂u P = ∂x v ∂v ∂y ∂u ∂y ∂v ∂z ∂u ∂z ∂v
(3.4) The twist vector is the rate of change of the tangent vector Pu with respect to v or Pv with respect to u, or it is the cross (mixed) derivative vector at the point. The twist vector can be written in terms of its Cartesian components as
The surface normal at a point is a vector that is perpendicular to both tangent vectors at the point, that is: N(u, v)= ∂P × ∂P = Pu × Pv (3.6) ∂u ∂v The unit normal vector is
The minimum length between two points is
S = ∫ ds = ∫
ta
tb
ห้องสมุดไป่ตู้tb
ta
Eu ' +2Fu' v'+Gv' dt
3.2 Surface Representation Surfaces can be described by nonparametric or parametric equations. Figure 3. 3 shows parametric representation of a surface. The parametric equation of threethree-dimensional curved surface is
Surfaces are usually displayed with a mesh, the mesh size is m × n (for example) and it can be changed by users. Figure 3.2 shows a cylinder surface with 3 X 4 and 6 X 8 mesh size respectively. The finer mesh has a smoother contour but takes more CPU time.
Similar to curves, CAD/CAM systems provide designers with both analytic and synthetic surfaces entities. Analytic surface includes entities such as plane surface, ruled surface, surface of revolution and tabulated cylinder. Synthetic entities include the bicubic Hermite spline surface, B-spline Bsurface, Bezier patches, Coons patches etc. This chapter covers the mathematical representations of most popular surfaces.
Pu × Pv N n= = N Pu × Pv
(3.7)
Many surfaces used in engineering are curved surfaces. It is useful to find the minimum length between two points on a curved surface and the minimum surface curvature at a point on surface. The paths that have minimum lengths between two points on a curved surface are known as geodesics. Surface geodesics can provide optimized motion planning across a curved surface for numerical control machining.
∂2x Puv = ∂u∂v ∂ y ∂u∂v
2
∂ z ∂2x ∂2y ∂ 2z i+ j+ k = ∂u∂v ∂u∂v ∂u∂v ∂u∂v
2
T
u min ≤ u ≤ u max , v min ≤ v ≤ v max
(3.5)
The twist vector depends on both the surface geometry and its parameterization. Due to the latter dependency, interpreting the twist vector in geometrical terms may be misleading. Each rectangular surface has sixteen vectors and four boundary curves. These vectors are four position vectors (four corners), eight tangent vectors (two at each corner), and four twist vectors at corners.
P(u , v) = [x
y z ] = [x (u , v) y(u , v) z( x , y)] , u min ≤ u ≤ u max , v min ≤ v ≤ v max
T T
The parametric variable u and v are constrained to intervals bounded by minimum and maximum values. In most surfaces, the parametric variables u and v have the interval [0, 1]. A surface can also be divided into a number of topological patches. Figure 3.4 shows a two-patch twosurface where u and v values are between [0, 1].
The choice of the surface form depends on the application and manufacturing methods. All surfaces form must be easy to differentiate to determine the surface tangents, normals and curvatures. Polynomial functions are obvious choices, and for most practical application, cubic polynomials are sufficient. Polynomials of higher order have higher computation costs and may lead to unwanted oscillations.
Despite many advantages, surface models have few disadvantages. Surface models are generally more complex and require more CPU time and computer storage than wireframe models. They can be ambiguous in the case when determining which surfaces define a volume.
Chapter 3 Surfaces
3.1 Introduction 3.2 Surface Representation 3.3 Parametric Representation of Analytical Surfaces 3.4 Parametric Representation of Synthetic Surfaces 3.5 Surface Manipulation
Curves on the surfaces can be generated by fixing one parametric variable, say u, to obtain a curve in terms of variable v. The mesh size determines the number of curves displayed. There are two tangent vectors at any point P(u, v) on the surface. These vectors are given by