CHAPTER 4
208
Graphics
PDF represents coordinates in a two-dimensional space. The point (x,
y)
in such
a space can be expressed in vector form as
[
x y
1 ]
. The constant third element of
this vector (
1
) is needed so that the vector can be used with 3-by-3 matrices in the
calculations described below.
The transformation between two coordinate systems is represented by a 3-by-3
transformation matrix written as follows:
a b
0
c d
0
e f
1
Because a transformation matrix has only six elements that can be changed, it is
usually specified in PDF as the six-element array
[
a b c d e f
]
.
Coordinate transformations are expressed as matrix multiplications:
a b
0
c d
0
e f
1
[
x′ y′
1
]
=
[
x y
1
] ×
Because PDF transformation matrices specify the conversion from the trans-
formed coordinate system to the original (untransformed) coordinate system,
x
′
and
y
′
in this equation are the coordinates in the untransformed coordinate sys-
tem, and
x
and
y
are the coordinates in the transformed system. The multiplica-
tion is carried out as follows:
x′
=
a
×
x
+
c
×
y
+
e
y′
=
b
×
x
+
d
×
y
+
f
If a series of transformations is carried out, the matrices representing each of the
individual transformations can be multiplied together to produce a single equiva-
lent matrix representing the composite transformation.
Matrix multiplication is not commutative—the order in which matrices are mul-
tiplied is significant. Consider a sequence of two transformations: a scaling trans-
formation applied to the user space coordinate system, followed by a conversion
from the resulting scaled user space to device space. Let
M
S
be the matrix specify-
ing the scaling and
M
C
the current transformation matrix, which transforms user