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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, while x and y are the coordinates in the transformed system Carrying out the multiplication, we have 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 signi cant 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 speci ed in the transformed space, the correct order of transformations must rst 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 ) where XD denotes the coordinates in device space XU denotes the coordinates in default user space XS denotes the coordinates in scaled user space



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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 = M T M Note: When rendering graphics objects, it is sometimes necessary for a viewer application to perform the inverse of a transformation that is, to nd 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

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component to invoke other Web resources are described in the section Invoking Other Web Resources, page (235)

A PDF viewer application maintains an internal data structure called the graphics state that holds current graphics control parameters These parameters de ne the global framework within which the graphics operators execute For example, the f ( ll) operator implicitly uses the current color parameter, and the S (stroke) operator additionally uses the current line width parameter from the graphics state The graphics state is initialized at the beginning of each page, using the default values speci ed in Tables 42 and 43 Table 42 lists those graphics state parameters that are device-independent and are appropriate to specify in page descriptions The parameters listed in Table 43 control details of the rendering (scan conversion) process and are device-dependent; a page description that is intended to be device-independent should not modify these parameters

Collaborating Web components share information via objects maintained as attributes of four scope objects These attributes are accessed with the methods of the class representing the scope Table 10 4 lists the scope objects Figure 10 1 shows the scoped attributes maintained by the Duke s Bookstore application Table 10 4 Scope Objects

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