
578 Chapter 12 Surfaces
2v
2(– u + w)
2v
2(– v + w)
–2w
0
–2v
–2w
u
2
2uv
2uw
2u
–2u
v
2
2vw
w
2
2u
0
Coefficients of x Coefficients of x
u
Coefficients of x
v
Figure 12.1 Polynomial coefficients for n = 2.
12.3 B´ezier Triangle Patches
B
´
ezier triangle patches are slightly more complicated to use than B
´
ezier rectang le
patches, but they are useful for creating models of arbitrary complexity.
12.3.1 Definitions
Given a triang le lattice of 3D control points P
i
0
, i
1
, i
2
for i
0
≥ 0, i
1
≥ 0, i
2
≥ 0, and
i
0
+ i
1
+ i
2
= n, the B
´
ezier triangle patch for the points is
X(u, v, w) =
|I |=n
B
n, I
(u, v, w) P
I
where I = (i
0
, i
1
, i
2
), |I |=i
0
+ i
1
+ i
2
, u ≥0, v ≥ 0, w ≥ 0, and u +v +w = 1. The
summation ...