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
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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