Computer Aided Engineering Design

DESIGN OF SURFACES 215

Example 7.4. The control points for a bi-cubic Bézier surface are given by

rrrr

rrrr

rrrr

rrrr

00 10 20 30

01 11 21 31

02 12 22 32

03 13 23 33

= {0, 0, 0} = {1, 0, 1] = {2, 0, 1} = {3, 0, 0}

= {0, 1, 1} = {1, 1, 2} = {2, 1, 2} = {3, 1, 1}

= {0, 2, 1} = {1, 2, 2} = {2, 2, 2} = {3, 2, 1}

= {0, 3, 0} = {1, 3, 1} = {2, 3, 1} = {3, 3, 0}

Plot the bi-cubic surface.
The equation for the bi-cubic Bézier surface patch is given in (7.28) and the parent surface is
shown in Figure 7.8 (a). As an exercise, we may determine the tangents, normal, and Gaussian and
mean curvatures at u = 0.5, v = 0.7. The effect of relocating control points is shown in Figures 7.8
(b and c). For new control points r 11 = {2, 2, 6} and r 21 = {4, 2, 6} we get Figure 7.8(b) and for
r 12 = {4, 6, 4} and r 22 = {4, 4, 4}, Figure 7.8(c) is obtained.

Figure 7.8 (a) Bézier bi-cubic patch in Example 7.4, (b) and (c) patches with data points relocated.

2

1.5

1

0.5

0

z

4

2

0

0 1

2 3

y 4

x

6

4

2

0

z

(^43)
(^21)
0 0
1
2
3
4 y 4 3 2 1 0
(^64)
2 0 0
2
4
z
y x
(c)
(a) (b)
x
Example 7.5. Triangular Bi-Cubic Bézier Patch. Collapsing the data points for any boundary curve
can create a triangular bi-cubic Bézier surface patch. Create the surface patch with the following
control points: