Graphics Reference
In-Depth Information
The bilinear surface is a special case of a lofted surface and also of the Coons
surface and the Bézier tensor product surface described later on. We can rewrite the
function p(u,v) in matrix form as
pp
pp
1
-
v
)
Ê
Ë
ˆ
¯
Ê
Ë
ˆ
¯
00
01
()
=-
(
puv
,
1
uu
.
v
10
11
If the points
p
i
are coplanar and linearly independent (as in Figure 12.10(b)), then the
resulting surface is a quadrilateral, otherwise (as in Figure 12.10(c)) it is a surface of
degree two, meaning that it is parameterized by quadratic polynomials.
12.6.1 Example.
If
p
00
= (1,0,0),
p
01
= (1,3,-1),
p
10
= (-1,-2,2), and
p
11
= (-1,1,1),
then it is easy to check that
(
)
=-
(
)
puv
,
123 22
u v
,
-
u u
,
-
v
.
Note that the points all lie on the plane 2x + y + 3z =1.
12.6.2 Example.
If we use the same points as in Example 12.6.1 but replace
p
00
by
p
00
= (0,0,0), then
(
)
=--
(
)
puv
,
v
u uv v
,
322
-
u u
,
-
v
.
The points are no longer coplanar and we see a quadratic term “uv” in the formula
for p(u,v).
A special case of the four-point interpolating surface arises when two of the points
are the same. We are then simply parameterizing a triangular planar patch. However,
it makes more sense in that case to use a triangular domain. We can do that if we use
barycentric coordinates. Let
p
0
,
p
1
, and
p
2
be three points and define
(
)
=++--
(
)
puv
,
u
pp
v
1
u
v
p
,
(12.18)
0
1
2
where 0 £ u,v and 0 £ u + v £ 1. The function p(u,v) parameterizes the triangle
p
0
p
1
p
2
.
If we drop all the constraints on u and v, then we get a parameterization of the plane
containing the three points called the
interpolating plane
. Note that equation (12.18)
can be rewritten in the form
(
)
=+
puv
,
ppppp
2
u
+
v
2
0
2
0
which shows more clearly that we are parameterizing a plane that has the vectors
p
2
p
0
and
p
2
p
0
for a basis.
12.7
Coons Surfaces
The surfaces described so far had a relatively simple description by means of a single
formula. There are many other surfaces that one needs to deal with in CAGD, such
