Graphics Reference
In-Depth Information
Now, say we keep the parameter u constant u
=
c:
=
=
+
+
z cv
k
Then
r
is a function of v.
r
v measures the rate of change of
r
relative to v and is represented
by a tangential vector, as shown in Fig. 8.4. Similarly, by keeping the parameter v constant
v
r
r
cv
x cv
i
y cv
j
=
c:
z uc
k
Then
r
is a function of u.
r
u measures the rate of change of
r
relative to u and is represented
by a tangential vector, as shown in Fig. 8.4.
r
=
r
uc
=
x uc
i
+
y uc
j
+
Consequently, the two vectors
r
u and
r
v identify a tangential plane at the point
r
uv, which enables us to compute the normal vector
n
:
r
u
×
r
v
n
=
(8.8)
However, we must remember that
n
could also be defined as
r
u
as there is no right or wrong side to a surface patch. It is up to us to determine which side we
wish represents the outside or inside.
r
v
×
n
=
Example 1
Consider, then, a quadratic Bézier patch defined as
⎡
⎤
⎡
⎤
v
2
p
00
p
01
p
02
p
10
p
11
p
12
p
20
p
21
p
22
−
1
=
1
u u
2
⎣
⎦
⎣
⎦
u
2
r
uv
−
2u1
−
−
2v 1
v
v
2
where
p
00
=
000
p
01
=
101
p
02
=
200
p
10
=
−
111
p
11
=
113
p
12
=
311
p
20
=
020
p
21
=
131
p
22
=
220
⎡
⎤
⎡
⎤
p
00
p
01
p
02
p
10
p
11
p
12
p
20
p
21
p
22
1
−
2v
+
v
2
=
1
u
2
2u
2u
2
u
2
⎣
⎦
⎣
⎦
r
uv
−
2u
+
−
2v
−
2v
2
v
2
⎡
⎤
⎡
⎤
012
−
1
−
2v
+
v
2
=
1
u
2
2u
2u
2
u
2
⎣
⎦
⎣
⎦
x
r
uv
−
2u
+
−
113
012
2v
−
2v
2
v
2
⎡
⎤
⎡
⎤
⎡
⎤
u
2
2u
2
v
2
1
−
2u
+
+
2
−
4u
+
+
1
−
2v
+
=
−
2u
2u
2
⎣
⎦
⎣
⎦
⎣
⎦
2u
2
6u
2
2v
2
x
r
uv
−
2u
−
+
6u
−
+
2v
−
u
2
2u
2
v
2
⎡
⎤
v
2
1
−
2v
+
=
2u
2
2u
1
−
2
⎣
⎦
2u
2
2v
2
v
2
x
r
uv
−
+
2u
+
2v
−
