Graphics Reference
In-Depth Information
Because the conventions are equivalent, it is immaterial which is used as long as consistency is
maintained. The 4
4 transformation matrix used in one of the notations is the transpose of the
4
4 transformation matrix used in the other notation.
2.1.3
Concatenating transformations: multiplying transformation matrices
One of the main advantages of representing basic transformations as square matrices is that they can be
multiplied together, which concatenates the transformations and produces a compound transforma-
tion. This enables a series of transformations,
M
i
, to be premultiplied so that a single compound
transformation matrix,
M
, can be applied to a point
P
(see
Eq. 2.7
). This is especially useful (i.e., com-
putationally efficient) when applying the same series of transformations to a multitude of points. Note
that matrix multiplication is associative ((
AB
)
C ¼ A
(
BC
)) but not commutative (
AB 6¼ BA
).
0
P
¼ M
1
M
2
M
3
M
4
M
5
M
6
P
M ¼ M
1
M
2
M
3
M
4
M
5
M
6
P
(2.7)
0
¼ MP
When using the convention of postmultiplying a point represented by a row vector by the same series of
transformations used when premultiplying a column vector, the matrices will appear in reverse order in
addition to being the transposition of the matrices used in the premultiplication.
Equation 2.8
shows the
same computation as
Equation 2.7
,
except in
Equation 2.8
,
a row vector is postmultiplied by the trans-
formation matrices. The matrices in
Equation 2.8
are the same as those in
Equation 2.7
but are now
transposed and in reverse order. The transformed point is the same in both equations, with the exception
that it appears as a column vector in
Equation 2.7
and as a row vector in
Equation 2.8
. In the remainder of
this topic, such equations will be in the form shown in
Equation 2.7
.
0
T
¼ P
T
M
6
M
5
M
4
M
3
M
2
M
1
M
T
¼ M
6
M
5
M
4
M
3
M
2
M
1
P
P
(2.8)
0
T
¼ P
T
M
T
2.1.4
Basic transformations
For now, only the basic transformations rotate, translate, and scale (uniform scale as well as nonuni-
form scale) will be considered. These transformations, and any combination of these, are
affine trans-
formations
[
4
]. It should be noted that the transformation matrices are the same whether the space is
left- or right-handed. The perspective transformation is discussed later. Restricting discussion to the
basic transformations allows the fourth element of each point vector to be assigned the value one
2
3
2
4
3
5
2
4
3
5
0
abcd
efgh
i j km
0001
x
y
1
x
4
5
¼
0
y
(2.9)
0
z
1
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