Computer Aided Engineering Design

202 COMPUTER AIDED ENGINEERING DESIGN

In Chapter 6, differential properties of surface patches are discussed using analytical surfaces.
They include the plane, ruled or lofted patches, surfaces of revolution, and sweep patches. These
patches are described in parametric form with examples of the mathematical background required for
the design of synthetic surfaces. In surface design, free form surfaces are created using surface
patches. The following gives a broad classification of surface patches

(a) Parametric polynomial patches (or tensor product surfaces)
(b) Boundary interpolating surfaces
(c) Sweep (linear or rotational) surface patches
(d) Quadric surface patches
Surface patches are bi-parametric, and the curve models developed in Chapters 4 and 5 are
directly extendible to their design, that is, Hermite, Bézier or B-spline surface patches can be created
using the basis functions for the respective curves described in these chapters.

7.1 Tensor Product Surface Patch

GivenΦ and Ψ as two sets of univariate functions such that

Φφ = i( )i=0, = Ψψ( ) =0

m

j j

n

{}u {}v (7.2)

with interval domains u∈U and v∈V, a surface

rC(, ) = () ()

=0 =0

uu

j

n

i

m

vvΣΣijφψi j (7.3)

is called a tensor product surfacewith domain U×V. The surface is bi-quadraticform = n = 2 and
bi-cubic for m = n = 3.

Example 7.1.Consider the first and second order Bézier basis functions

Φ(u) = {φ 0 (u)φ 1 (u)} = {(1 –u)u},

Ψ(v) = {ψ 0 (v)ψ 1 (v)ψ 2 (v)} = {(1 –v)^2  2 v(1 –v) v^2 }

The equation of the tensor product surface is given by

r(u,v) = C 00 φ 0 ψ 0 + C 01 φ 0 ψ 1 + C 02 φ 0 ψ 2 + C 10 φ 1 ψ 0 + C 11 φ 1 ψ 1 + C 12 φ 1 ψ 2

GivenCij as C 00 = [0 0 0], C 10 = [1 2 0], C 01 = [0 2 4], C 11 = [1 2 4] C 02 = [0 –1 3], C 12 = [1 –1 3],
the equation of the tensor product surface may be written in the following form where the ordered
triple [xyz] is a function of parameters u and v.

r(u,v) = [x y z] = [(1 –u) u]

(0, 0, 0) (0, 2, 4) (0, –1, 3)

(1, 2, 0) (1, 2, 4) (1, –1, 3)

(1 – )

2 (1 – )

2

2

⎡

⎣

⎢

⎤

⎦

⎥

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

v

vv

v

(7.4)

The surface generated is shown in Figure 7.1. The thick lines represent v= constant values and the
thick curve represents u= constant values on the surface.
We can generalize the form for a tensor product surface as