Graphics Reference
In-Depth Information
for i = 0 or 1.
Proof.
Exercise 12.7.2.
As a simple application of Theorem 12.7.1 we can get a smoother Coons surface
by using the Hermite basis functions H
i,3
introduced in equation (11.18) of Section
11.2.2 and defining
-
( )
-
() ( )
-
()
-
()
()
()
p
00
,
p
01
,
p
0
,
v
Hv
Hv
Ê
ˆ
Ê
ˆ
03
,
Á
Á
˜
˜
Á
Á
˜
˜
(
)
=
(
()
()
)
( )
puv
,
H
u H
u
1
p
10
,
p
11
,
p
1
,
v
.
(12.23)
03
,
33
,
33
,
Ë
(
)
(
)
¯
Ë
¯
pu
,
0
pu
,
1
0
1
All three conditions of Theorem 12.7.1 are satisfied by H
0,3
and H
3,3
.
Definition.
The parametric surface defined by the function p(u,v) in equation (12.23)
is called the
bicubic Coons patch
or
surface
for the curves p(0,v), p(1,v), p(u,0), and
p(u,1).
One of the nice properties of the bicubic Coons surface is that it gives us smooth
global surfaces. Specifically, given a network of curves, the global interpolating surface
that one would get using the basic bilinearly blended Coons parameterization for each
individual patch would be only C
0
even if the curves themselves are C
1
. This problem
is caused by the use of linear blending functions. If we use the bicubic Coons surface,
then we get a globally C
1
surface. This follows from Theorem 12.7.1 and properties
of the functions H
0,3
and H
3,3
.
Another interesting fact about the surface (Pp)(u,v) defined by equations (12.22)
is that if
()
+
()
=
()
+
()
= ,
bu bu
1
or cv cv
1
(12.24)
0
1
0
1
then it is
affinely invariant
in the sense that if the surface is transformed by an affine
transformation, then the transformed version can be computed from equations (12.22)
applied to the transformed boundary data. By a simple extension of Theorem 11.2.2.3
we only have to show that the coefficients of equation (12.22b) add to 1, that is,
()
+
()
+
()
+
()
-
() ()
-
() ()
-
() ()
-
() ()
= ,
bu bu cv cv bucv bucv bucv bucv
0
1
1
0
1
0
0
0
1
1
0
1
1
but this equation can be rewritten in the form
(
()
+
()
-
)
(
( )
+
()
-
)
=
bu bu
1
cv cv
1
0
.
0
1
0
1
It follows that both the bilinearly blended and bicubic Coons surface are affinely
invariant.
Even though the bicubic Coons patch gives us a C
1
surface if the boundary curves
are C
1
, we have little control over the derivatives along the boundaries. To get more
flexibility assume that we are also given the partial derivatives p
u
(0,v), p
u
(1,v), p
v
(u,0),
and p
v
(u,1). See Figure 12.13. Define new operators Q
1
and Q
2
by
