Graphics Reference
In-Depth Information
Table 9.5.
The x, y, z coordinates for different
values of u and v
V
1
2
0
1
1
,
1
2
,
0
0
(0
,
0
,
0)
(2
,
0
,
0)
0
,
1
2
,
1
1
,
1
2
,
1
2
,
1
2
,
1
1
2
u
1
,
1
2
,
2
1
(0
,
0
,
2)
(2
,
0
,
2)
Table 9.5 shows the coordinate values for different values of
u
and
v
.Inthis
example, the
y
-coordinates provide the surface curvature, which could be
enhanced by modifying the
y
-coordinates of the control points.
9.7.3 Cubic Bezier Surface Patch
As we saw earlier in this chapter, cubic Bezier curves require two end-points,
and two central control points. In the surface patch formulation a 4
×
4 matrix
is required as follows:
⎡
⎤
⎡
⎤
−
13
−
31
P
00
P
01
P
02
P
03
⎣
⎦
·
⎣
⎦
·
3
−
630
P
10
P
11
P
12
P
13
P
uv
=[
u
3
u
2
u
1]
·
−
3300
1000
P
20
P
21
P
22
P
23
P
30
P
31
P
32
P
33
⎡
⎣
⎤
⎦
·
⎡
⎣
⎤
⎦
v
3
v
2
v
1
−
13
−
31
3
−
630
−
3300
1000
which can be illustrated by an example.
Given the points:
P
00
=(0
,
0
,
0)
P
01
=(1
,
1
,
0)
P
02
=(2
,
1
,
0)
P
03
=(3
,
0
,
0)
P
10
=(0
,
1
,
1)
P
11
=(1
,
2
,
1)
P
12
=(2
,
2
,
1)
P
13
=(3
,
1
,
1)
P
20
=(0
,
1
,
2)
P
21
=(1
,
2
,
2)
P
22
=(2
,
2
,
2)
P
23
=(3
,
1
,
2)
P
30
=(0
,
0
,
3)
P
31
=(1
,
1
,
3)
P
32
=(2
,
1
,
3)
P
33
=(3
,
0
,
3)