
11.6 NURBS Curves 561
(x(u), y(u)) =
w
0
(1 − u)
2
(1, 0) + w
1
2u(1 − u)(1, 1 ) + w
2
u
2
(0, 1)
w
0
(1 − u)
2
+ w
1
2u(1 − u) + w
2
u
2
for u ∈ [0, 1]. The requirement that x
2
+y
2
=1leads to the weights constraint 2w
2
1
=
w
0
w
2
. The choice of weights w
0
= 1, w
1
= 1, and w
2
= 2 leads to a well-known
parameterization:
(x(u), y(u)) =
(1 − u
2
,2u)
1 + u
2
If you were to tessellate the curve with an odd number of uniform samples of u,say,
u
i
= i/(2n) for 0 ≤ i ≤ 2n, then the resulting polyline is not symmetric about the
midpoint u = 1/2. To obtain a sy mmetric tessellation, you need to choose w
0
= w
2
.
The weight constraint then implies w
0
= w
1
√
2. The parameterization is then
(x(u), y(u))