Graphics Reference
In-Depth Information
straints that define the parameterization p(u,v) completely. We shall derive a formula
for p(u,v) in the uniform spacing case, but rather than finding the algebraic coeffi-
cients by solving equations, we shall use a matrix approach similar to what we did in
the case of the four-point problem for curves. We start with the equation (12.36) and
seek a matrix
M
l
so that
(
)
=
UM PM
T
T
puv
,
V
,
(12.39)
1
1
where
P
= (
p
ij
= p(i/3, j/3) ). It is easy to check that the matrix
M
l
, which solves (12.39)
is the same four-point matrix
M
l
that was defined in equation (11.43), that is,
9
2
27
2
27
2
9
2
Ê
ˆ
-
-
Á
Á
Á
Á
Á
Á
˜
˜
˜
˜
˜
˜
45
2
9
2
9
-
18
-
M
1
=
.
11
2
9
2
-
9
-
1
Ë
¯
1
0
0
0
Furthermore, the
B
and
P
matrices are related by the equation
B
=
LPL
T
, where the
matrix
L
is as in equation (11.44).
Up to this point we have assumed that the domain for our bicubic patch is [0,1]
¥ [0,1], but what if we were to reparameterize and use a different domain [a,b] ¥ [c,d]?
The answer is similar to the answer we gave in Section 11.3 for cubic curves. The geo-
metric matrix would have to be defined in terms of the values and derivatives of the
function at the endpoints a, b, c, and d.
12.10
Bézier Surfaces
Defining tangents and especially the twist vectors for the bicubic patch is not very
intuitive for many users. Again, a Bézier approach can be followed, but this time we
specify a grid of sixteen points
p
ij
. See Figure 12.17. Let
pppp
pppp
pppp
pppp
Ê
ˆ
00
01
02
03
Á
Á
Á
˜
˜
˜
10
11
12
13
B
b
=
(12.40)
20
21
22
23
Ë
¯
30
31
32
33
and define a parameterization p(u,v), 0 £ u,v £ 1 , by
(
)
=
UM B M
bb b
T
T
puv
,
V
,
(12.41)
where
M
b
is the matrix defined in (11.47).
