Mathematical Prerequisites - Introduction to 3D Game Programming with DirectX 9.0 - page 24

1000
0100

T 1, 2 , 3

0010
12 31

Applying the sequence of transformations in the order scaling, rotating,
and translating, we obtain:

pS 1, 0, 0, 1 p

(5)

pR .707, 0, .707, 1 p

y

pT 1.707, 2, 3.707, 1

One of the key benefits of matrices is that we can use matrix multipli-
cation to combine several transformations into one matrix. For
example, let's reconsider the example at the beginning of this section.
Let's combine the three transformation matrices into one matrix repre-
senting all three through matrix multiplication. Note that the order in
which we multiply the transformations is the order that they are
applied.

1

0 0 0

. 707 0 . 707 0 1 0 0 0

5

1

0 1 0 0 0 1 0 0

0 0 0

SR T

5

y

. 707 0 . 707 0 0 0 1 0

1

0 0 0

0 0 0 1 1 2 3 1

5

(6)

0 0 0 1

. 1414 0 . 1414 0

0 1 0 0

Q

. 1414 0 . 1414 0

1 2 3 1

Then pQ = [1.707, 2, -3.707, 1].
The ability to combine transformations has performance implica-
tions. Suppose that we need to apply the same scaling, rotation, and
translation transformations to a large set of vectors (a common task in
3D graphics). Instead of applying a sequence of transformations, as we
did in equation (5), per vector, we can combine all three transforma-
tions into one matrix, as we did in equation (6). Then we only have to

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