Represent a point by the column of homogeneous coordinates . The actions of the four matrices are:
MatrixGeometric action and parameters
A proper rotation about the -axis: . The angle is in radians, with the usual right-hand sense.
A translation by : .
Uniform scaling by : . Positive gives a uniform dilation; collapses the object to the origin; negative additionally reverses its direction through the origin.
Perspective projection from the origin onto the plane , where is the focal length in the chosen coordinate units.
The last operation produces . It requires a perspective divide: divide the first three components by the last. For ,
(1)
This is why the last component of homogeneous coordinates cannot simply be discarded. Points with have no finite projected position, and a visible scene normally uses under this forward-axis convention. The first three transformations leave the last component equal to one and need no nontrivial divide. The displayed retains no depth variation after division; it describes projection onto the image plane, rather than a complete depth-buffer mapping.

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