Graphics Reference
In-Depth Information
Definition.
The parametric surface p(u,v) defined by equation (12.43) is called the
Bézier surface
defined by the
control points
p
ij
.
The points of a Bézier surface are efficiently computed by the two-dimensional
analog of the de Casteljau algorithm (see [PieT95]). Partial derivatives are easily deter-
mined since we already know the derivatives for Bézier curves (see equation (11.57)):
n
m
∂
∂
p
u
Ê
Á
∂
∂
ˆ
˜
Â
Â
(
)
=
()
()
uv
,
B
v
BuP
jn
,
im
,
ij
u
j
=
0
i
=
0
n
m
-
1
Â
Â
=
mB
()
u B
()
v
(
1
pp
-
)
.
im
,
-
1
jn
,
i
+
,
j
ij
j
=
0
i
=
0
Similarly,
m
n
-
1
∂
∂
p
v
Â
Â
(
)
=
()
()
(
)
uv
,
n
B
uB
v
pp
-
.
im
,
jn
,
-
1
i j
,
+
ij
i
=
0
j
=
0
Formulas for higher derivatives are obtained just like in the case of curves.
Here are some properties of a Bézier surface, many of which are similar to those
of Bézier curves:
(1) The boundary curves of a Bézier surface are Bézier curves.
(2) Only the corner vertices are interpolated but the shape of the surface closely
follows the control points
p
ij
.
(3) The vectors
p
00
p
10
and
p
00
p
01
generate the tangent plane at
p
00
, with similar
facts at the other corner points.
(4) The Bézier patch lies in the convex hull of its control points.
12.10.1
Theorem.
The Bézier surface defined by (12.42) is affinely invariant.
Proof.
This follows from the Theorem 11.2.2.3.
Bézier surfaces do not satisfy any known variation diminishing property (see
[PraG92]).
Just like in the case of Bézier curves, one drawback with Bézier surfaces is that
the degree of the surface increases as the number of control points increases. One can
counter this problem by restricting oneself to cubic patches. See Section 12.15 for
ways to ensure that patches meet smoothly.
12.11
Gregory Patches
Gregory patches are an extension of the Coons patch and Bézier surface. Chiyokura
([Chiy88]) has an extensive discussion of the surface and its uses. See also [HosL93]
or [Salo99]. Rather than using 16 control points for a patch like an ordinary
