Graphics Reference
In-Depth Information
(
)
(
)
(
)
(
)
p
00
,
p
01
,
p
00
,
p
01
,
Ê
ˆ
v
v
Á
Á
Á
˜
˜
˜
()
()
()
()
p
10
,
p
11
,
p
10
,
p
11
,
v
v
B
=
,
(12.28)
(
)
(
)
(
)
(
)
p
00
,
p
01
,
p
00
,
p
01
,
u
u
uv
uv
Ë
¯
()
()
()
()
p
10
,
p
11
,
p
10
,
p
11
,
u
u
uv
uv
then equation (12.26) can be written in the form
( ( )
=
Qp u v
,
(
)
()
()
()
pv
0
1
0
1
,
,
cv
cv
ev
e
Ê
ˆ
Ê
ˆ
0
Á
Á
Á
Á
Á
˜
˜
˜
˜
˜
Á
Á
Á
Á
Á
˜
˜
˜
˜
˜
()
( )
()
-
B
p v
pv
pv
1
(
() ()
() ()
)
bubududu
1
,
,
.
(12.29)
0
1
0
1
u
0
()
u
1
1
Ë
¯
Ë
¯
(
)
(
)
(
)
(
)
pu
,
0
pu
,
1
p u
,
0
p u
,
1
0
v
v
The surface Qp(u,v) defined by equation (12.29) will interpolate the boundary curves
p(0,v), p(1,v), p(u,0), p(u,1), their derivatives p
u
(0,v), p
u
(1,v), p
v
(u,0), p
v
(u,1), and the
values p
uv
(0,0), p
uv
(0,1), p
uv
(1,0), p
uv
(1,1). Using such parameterizations we can now
define a network of patches from a given grid of boundary curves p(i,v) and p(u,j), i,j
= 1,2,..., as shown in Figure 12.14 that is a globally C
1
surface. It is also affinely
invariant provided that equations (12.24) are satisfied. Section 12.9 will have more to
say about the B matrix in equation (12.28). We summarize our results about networks
of Coons surfaces in Table 12.7.1.
Figure 12.14.
A smooth network of Coons
surfaces.
T
able 12.7.1
Properties of Coons networks.
Individual patch p(u,v)
Continuity of
of Coons networks
network
Comments
C
0
Bilinearly blended (equations (12.20))
Affinely invariant
C
1
Bicubic (equation (12.23))
Affinely invariant
C
1
Beneralized bicubic (equation (12.26))
Affinely invariant, allows control over
partials along boundary
