Graphics Reference
In-Depth Information
Figure 12.12.
A bilinearly blended Coons patch.
Interpolating this error term in the v-direction is just the function P
2
(p - P
1
p)(u,v). If
we were to arbitrarily define
(
)
=
(
)
+
(
)(
)
=
(
)
+
(
)
-
(
)
puv
,
Ppuv
,
P p Pp uv
-
,
Ppuv
,
Ppuv
,
PPpuv
,
,
(12.20a)
1
2
1
1
2
2 1
then by construction this function p(u,v) would now interpolate our
entire
boundary.
What would have happened if we had applied this argument to P
2
p and its difference
with the actual values of p along the v-direction boundaries? Fortunately, because the
operators P
1
and P
2
commute, that is, P
2
P
1
= P
1
P
2
, we would have arrived at the
same
formula (12.20a). As one can see from Figure 12.12, P
1
P
2
p(u,v) is what is called a
doubly ruled surface. Replacing the operators P
i
in the formula for p(u,v) in equation
(12.20a) by their definitions leads to the original Coons formula
(
)
=-
(
) (
)
+
(
+-
(
) (
)
+
()
p u v
,
1
v p u
,
0
vp u
,
1
1
u p
0
,
v
up
1
,
v
(
)
(
) (
)
--
(
)
(
)
--
(
) (
)
-
()
--
11 001
u
-
v p
,
u vp
,
11
u
v p
,
0
uvp
,
1
.
(12.20b)
Definition.
The parametric surface defined by the function p(u,v) in equations
(12.20) is called the (
bilinearly blended
)
Coons patch
or
Coons surface
for the curves
p(0,v), p(1,v), p(u,0), and p(u,1).
The Coons surface can be expressed in matrix form as follows:
( )
()
( )
()
( ) ( )
() ()
pv
pv
0
1
,
,
pu
pu
,
,
0
1
p
00
,
p
01
,
1
-
v
Ê
Ë
ˆ
¯
+-
Ê
Ë
ˆ
¯
Ê
Ë
ˆ
¯
Ê
Ë
ˆ
¯
(
)
=-
(
)
(
)
(
)
puv
,
1
uu
1
vv
--
1
uu
(12.20c)
p
10
,
p
11
,
v
-
( )
-
() ( )
-
()
-
()
p
00
,
p
01
,
p
0
,
v
1
-
v
Ê
ˆ
Ê
ˆ
Á
Á
˜
˜
Á
Á
˜
˜
(
)
( )
.
=-
1
uu
1
p
10
,
p
11
,
p
1
,
v
v
(12.20d)
Ë
(
)
(
)
¯
Ë
¯
pu
,
0
pu
,
1
0
1
The basic Coons surface is easily generalized by replacing the simple linear
blending functions by others. Let b
0
(t), b
1
(t), c
0
(t), and c
1
(t) be arbitrary real-valued
functions on [0,1] and define new operators P
1
and P
2
on functions p(u,v) by
