Editing Transformation matrix (section)

===Perspective projection===
{{main|Perspective projection}}
{{further|Pinhole camera model}}
[[File:Perspective transformation matrix 2D.svg|thumb|Comparison of the effects of applying 2D affine and perspective transformation matrices on a unit square.]]

Another type of transformation, of importance in [[3D computer graphics]], is the [[perspective projection]].  Whereas parallel projections are used to project points onto the image plane along parallel lines, the perspective projection projects points onto the image plane along lines that emanate from a single point, called the center of projection.  This means that an object has a smaller projection when it is far away from the center of projection and a larger projection when it is closer (see also [[Multiplicative inverse|reciprocal function]]).

The simplest perspective projection uses the origin as the center of projection, and the plane at <math>z = 1</math> as the image plane.  The functional form of this transformation is then <math>x' = x / z</math>; <math>y' = y / z</math>.  We can express this in [[homogeneous coordinates]] as:
<math display="block">\begin{bmatrix} x_c \\ y_c \\ z_c \\ w_c \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 &  1 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \\ 1 \end{bmatrix}=\begin{bmatrix} x \\ y \\ z \\ z \end{bmatrix}
</math>

After carrying out the [[matrix multiplication]], the homogeneous component <math>w_c</math> will be equal to the value of <math>z</math> and the other three will not change. Therefore, to map back into the real plane we must perform the '''homogeneous divide''' or '''perspective divide''' by dividing each component by <math>w_c</math>:
<math display="block">\begin{bmatrix} x' \\ y' \\ z' \\ 1 \end{bmatrix} = \frac{1}{w_c} \begin{bmatrix} x_c \\ y_c \\ z_c \\ w_c \end{bmatrix}=\begin{bmatrix} x / z \\ y / z \\ 1 \\ 1 \end{bmatrix}</math>

More complicated perspective projections can be composed by combining this one with rotations, scales, translations, and shears to move the image plane and center of projection wherever they are desired.