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w
P
=
xyzw
,,,
P
P
w =
1
y
x
Figure 4.5. A 4D point P is projected into three-dimensional space by calculating the
point where the line connecting the point to the origin intersects the space where
w
=
1
.
ourselves to 3
matrices in this section since tangent and normal directions are
unaffected by translations.) Some care must be taken when transforming normal
vectors, however. Figure 4.6 shows what can happen when a nonorthogonal ma-
trix M is used to transform a normal vector. The transformed normal can often
end up pointing in a direction that is not perpendicular to the transformed surface.
Since tangents and normals are perpendicular, the tangent vector T and the
normal vector N associated with a vertex must satisfy the equation
×
3
N T . We
must also require that this equation be satisfied by the transformed tangent vector
=
0
T and the transformed normal vector
N . Given a transformation matrix M , we
know that
TMT . We would like to find the transformation matrix G with
which the vector N should be transformed so that
′ =
(
) (
)
N
T NMT .
=
=
0
(4.30)
A little algebraic manipulation gives us
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