Game Development Reference
In-Depth Information
11. The result vector element w
i
is the product of the ith row of N multiplied by the column
vector v. To have w
i
=
i
j=1
v
j
, the ith row of N needs to capture all elements of v up
to and including the ith element, but exclude all others. This means that
1
if j ≤ i,
n
ij
=
0
otherwise.
Thus,
2
4
3
5
1
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
1
1
1
0
0
0
0
0
0
0
1
1
1
1
0
0
0
0
0
0
1
1
1
1
1
0
0
0
0
0
N =
.
1
1
1
1
1
1
0
0
0
0
1
1
1
1
1
1
1
0
0
0
1
1
1
1
1
1
1
1
0
0
1
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
12.
(a) Note that the structure of M causes the ith row of MN to be equivalent to the
difference between the ith and (i−1)th rows of N.
(b) Note that the structure of N causes the ith row of NM to be equivalent to the sum
of the first i rows of M.
2
4
3
5
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
(c) MN = NM = I
10×10
=
.
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
B.5
Chapter 5
(Page 159.)
1. Yes, any matrix expresses a linear transformation. Furthermore, because all linear trans-
formations are also a
ne transformations, the transform is also an a
ne transformation.
(There just isn't any translation in the a
ne transform, or equivalently, the translation
portion is zero.)
2
4
3
5
2
4
3
5
1
0
0
1.000
0.000
0.000
cos(−22
o
)
sin(−22
o
)
2.
0
=
0.000
0.927 −0.375
0 −sin(−22
o
)
cos(−22
o
)
0.000
0.375
0.927
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