Graphics Reference
In-Depth Information
Generalized Coons surfaces p(u,v) are nice because they are flexible. They inter-
polate arbitrary boundary curves and partials along the boundary. They have some
problems however. The first problem is that someone using them would also have to
specify p
uv
(0,0), p
uv
(0,1), p
uv
(1,0), and p
uv
(1,1). This is not easy because, since p(u,v)
is unknown, all one can do is estimate those unintuitive values. We describe one
way that avoids having to specify the mixed partials. Let a(v) = p
u
(0,v), b(v) = p
u
(1,v),
g(u) = p
v
(u,0), d(u) = p
v
(u,1). Replace the 2 ¥ 2 submatrix of vectors p
uv
(0,0), p
uv
(0,1),
p
uv
(1,0), and p
uv
(1,1) in the matrix B of equation (12.29) by the 2 ¥ 2 matrix
of parameterized vectors
u
d
dv
a
v
d
du
uv
g
u
d
dv
a
d
du
d
()
+
()
()
+-
(
)
( )
0
0
1
1
v
0
+
uv
+-
1
d
dv
b
v
d
du
uv
g
d
dv
b
d
du
d
(
)
( )
+
()
(
)
( )
+-
(
)
( )
1
-
u
0
1
1
-
u
1
1
v
1
(12.30)
.
1
-+
1
-+-
uv
1
Definition.
The parameterized surface that we get from this new data is called the
Gregory square
.
Note that it might have been tempting to have written expressions like
u
d
dv
a
()
as up
vu
(0,0), but we did not on purpose. It would have been confusing because
it would have looked as if we needed to specify the mixed second partial p
vu
(u,v)
at (0,0), which is not case. We computed the value from the curve a(v) that we
were given. Note also that the terms in matrix (12.30) are just convex combinations
of the derivatives of the curves a(v), b(v), g(u), and d(u) and that the mixed partials
that we get at the corners of the patch now depend on the direction in which we
approach the corner. See [Chiy88], [HosL93], or [Fari97] for more about this surface
patch.
A second problem for generalized Coons surfaces p(u,v) is that nice func-
tions, such as C
2
functions, have the property that one can interchange the order
of partial differentiation, but this may not be true for the parameterization p(u,v).
Specifically, ∂
2
p/∂u∂v may not equal ∂
2
p/∂v∂u at the corners of the patch. Achieving
equality of these two mixed partials is referred to as a
compatibility condition
.
(That term is used for other conditions such as having the boundary curves meet at
the corners or adjacent patches having the same tangent planes where they meet.) A
consequence of unequal mixed partials is that the projection operators Q
1
and Q
2
in equation (12.26) may not commute. The Gregory square does not satisfy the
compatibility condition. On the other hand, the Gregory patch discussed later in
Section 12.11 does.
Finally, Coons surfaces are a special case of
Gordon surfaces
. The latter inter-
polate a network of m curves in the u-direction and n curves in the v-direction. The
Coons surface is the case m = n = 1. It is also possible to define triangular Coons
patches. See [HosL93], [Fari97], or [Salo99].
