FS_MATRIX Class

• Transformations alter coordinate systems, not graphics objects. All objects painted before a transformation is applied are unaffected by the transformation. Objects 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 coordinates used after the transformation are expressed in the transformed coordinate 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 objects rather than of coordinate systems. Although either approach is correct and selfconsistent, some details of the calculations differ depending on which point of view is taken.

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|
            [ x′ y′ 1 ] =  [ x y 1 ] x |c d 0|
                                       |e f 1|
            

Because PDF transformation matrices specify the conversion from the transformed coordinate system to the original (untransformed) coordinate system, x′ and y′ in this equation are the coordinates in the untransformed coordinate system, and x and y are the coordinates in the transformed system. The multiplication 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 equivalent matrix representing the composite transformation

Matrix multiplication is not commutative — the order in which matrices are multiplied is significant. Consider a sequence of two transformations: a scaling transformation applied to the user space coordinate system, followed by a conversion from the resulting scaled user space to device space. Let Ms be the matrix specifying the scaling and Mc the current transformation matrix, which transforms user space to device space. Recalling that coordinates are always specified in the transformed space, the correct order of transformations must first convert the scaled coordinates to default user space and then the default user space coordinates to device space. This can be expressed as

            Xd = Xu × Mc  = (Xs × Ms) × Mc = Xs × (Ms × Mc)
            

            Xd denotes the coordinates in device space

            Xu denotes the coordinates in default user space

            Xs denotes the coordinates in scaled user space 
            

This shows that when a new transformation is concatenated with an existing one, the matrix representing it must be multiplied before (premultiplied with) the existing transformation matrix.

This result is true in general for PDF: when a sequence of transformations is carried out, the matrix representing the combined transformation (M′) is calculated by premultiplying the matrix representing the additional transformation (Mt) with the one representing all previously existing transformations (M):

            M′ = Mt × M
            

Note

When rendering graphics objects, it is sometimes necessary for an application to perform the inverse of a transformation—that is, to find the user space coordinates that correspond to a given pair of device space coordinates. Not all transformations are invertible, however. For example, if a matrix contains a, b, c, and d elements that are all zero, all user coordinates map to the same device coordinates and there is no unique inverse transformation. Such noninvertible transformations are not very useful and generally arise from unintended operations, such as scaling by 0. Use of a noninvertible matrix when painting graphics objects can result in unpredictable behavior.

Reference