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P
'
V
u
WC
P
'
P
'
OC
y
V
v
V
w
P
y
x
M
P
o
z
OC to WC
Transformation
P
z
Figure 16.17.
Transforming the anchoring positions of an orthonormal basis.
such that
⎧
⎨
P
o
=
P
o
M
,
P
y
=
P
y
M
,
.
⎩
P
z
=
P
z
M
.
Now, we can compute
V
y
P
y
−
P
o
,
=
V
z
P
z
−
P
o
,
=
and
V
z
.
In this way, the resulting orthonormal basis is
V
x
=
V
y
×
⎧
⎨
⎧
⎨
V
x
,
V
w
,
V
u
,
V
v
,
V
w
=
V
z
×
V
x
,
Orthonormal basis
=
where
V
u
=
⎩
⎩
V
z
.
=
V
v
Notice that we must explicitly normalize the vectors
V
w
,
V
u
,and
V
v
. List-
ing 16.22 shows the implementation that computes the orthonormal basis of the
Source file.
Model
_
Simulation.cpp
file in the
Model
folder of
the
D3D
_
SceneGraphIn3D
project.
shusui scene node. At label A, we gain access to the
M
t
and
M
s
transforms. At
label B, the general transformation
M
is computed by concatenating
M
t
and
M
s
transforms on the matrix stack. The orthonormal basis anchor positions
P
o
,
P
y
,
and
P
z
are then transformed. The vectors for the orthonormal basis (
V
w
,
V
u
,and
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