Graphics Reference
In-Depth Information
4
2
2 0
4
4
2
k twist factor
x x cos ( kz ) y sin( kz )
y
0
x sin ( kz )
y cos( kz )
z z
2
4
4
2
0
2
4
FIGURE 4.17
Twist about an axis.
s ¼ T U
ð
Þ P P 0
ð
Þ=
ð
ð
T U
ÞS
Þ
(4.3)
t ¼ U S
ð
Þ P P 0
ð
Þ= U S
ð
ð
ÞT
Þ
(4.4)
u ¼ S T
ð
Þ P P 0
ð
Þ= S T
ð
ð
ÞU
Þ
(4.5)
In these equations, the cross-product of two vectors forms a third vector that is orthogonal to the first
two. The denominator normalizes the value being computed. In the first equation, for example, the
projection of S onto TU determines the distance within which points will map into the range 0
1.
Given the local coordinates ( s , t, u) of a point and the unmodified local coordinate grid, a point's
position can be reconstructed in global space by simply moving in the direction of the local coordinate
axes according to its local coordinates ( Eq. 4.6 ):
<s<
P ¼ P 0 þ sS þ tT þ uU
(4.6)
To facilitate the modification of the local coordinate system, a grid of control points is created in the par-
allelepiped defined by the S, T , U axes. There can be an unequal number of points in the three directions. For
example, in Figure 4.21 , there are three in the S direction, two in the T direction, and one in the U direction.
If, not counting the origin, there are l points in the S direction, m points in the T direction, and n
points in the U direction, the control points are located according to Equation 4.7 .
0
0
i l
i
l S þ
j
m T þ
k
n U
@
P ijk ¼ P 0 þ
for
0
j m
(4.7)
0
k n
 
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