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SECTION 4.2 Coordinate Systems
A transformation matrix in PDF is specified by six numbers, usually in the form
of an array containing six elements. In its most general form, this array is denoted
[ a b c d e f ]; it can represent any linear transformation from one coordinate
system to another. This section lists the arrays that specify the most common
transformations; Section 4.2.3, “Transformation Matrices,” discusses more math-
ematical details of transformations, including information on specifying transfor-
mations that are combinations of those listed here:
• Translations are specified as [ 1 0 0 1 tx ty ], where tx and ty are the distances
to translate the origin of the coordinate system in the horizontal and vertical
dimensions, respectively.
• Scaling is obtained by [ sx 0 0 sy 0 0 ]. This scales the coordinates so that 1
unit in the horizontal and vertical dimensions of the new coordinate system is
the same size as sx and sy units, respectively, in the previous coordinate system.
• Rotations are produced by [ cos θ sin θ −sin θ cos θ 0 0 ], which has the effect
of rotating the coordinate system axes by an angle θ counterclockwise.
• Skew is specified by [ 1 tan α tan β 1 0 0 ], which skews the x axis by an angle
α and the y axis by an angle β.
Figure 4.5 shows examples of each transformation. The directions of translation,
rotation, and skew shown in the figure correspond to positive values of the array
elements.
sy b
ty
q
a
tx sx
Translation Scaling Rotation Skewing
FIGURE 4.5 Effects of coordinate transformations
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