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      SECTION 4.2                                                       Coordinate Systems



      The following transformations are shown in the figure:

      • A translation of 10 units in the x direction and 20 units in the y direction
      • A rotation of 30 degrees
      • A scaling by a factor of 3 in the x direction
      In the figure, the axes are shown with a dash pattern having a 2-unit dash and a
      2-unit gap. In addition, the original (untransformed) axes are shown in a lighter
      color for reference. Notice that the scale-rotate-translate ordering results in a
      distortion of the coordinate system, leaving the x and y axes no longer perpendic-
      ular; the recommended translate-rotate-scale ordering results in no distortion.


4.2.3 Transformation Matrices

      This section discusses the mathematics of transformation matrices. It is not
      necessary to read this section to use the transformations described previously;
      the information is presented for the benefit of readers who want to gain a deeper
      understanding of the theoretical basis of coordinate transformations.

      To understand the mathematics of coordinate transformations in PDF, it is vital
      to remember two points:

      • Transformations alter coordinate systems, not graphics objects. All objects paint-
        ed before a transformation is applied are unaffected by the transformation. Ob-
        jects painted after the transformation is applied are interpreted in the
        transformed coordinate system.
      • Transformation matrices specify the transformation from the new (transformed)
        coordinate system to the original (untransformed) coordinate system. All coor-
        dinates used after the transformation are expressed in the transformed coordi-
        nate system. PDF applies the transformation matrix to find the equivalent
        coordinates in the untransformed coordinate system.

      Note: Many computer graphics textbooks consider transformations of graphics ob-
      jects rather than of coordinate systems. Although either approach is correct and self-
      consistent, some details of the calculations differ depending on which point of view
      is taken.

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