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3D Game Engine Design, 2nd Edition
book

3D Game Engine Design, 2nd Edition

by David Eberly
November 2006
Intermediate to advanced content levelIntermediate to advanced
1040 pages
26h 31m
English
CRC Press
Content preview from 3D Game Engine Design, 2nd Edition
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))
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Publisher Resources

ISBN: 9781482267303