Editing Orthographic projection (section)

== Geometry ==
[[File:Graphical projection comparison.png|thumb|right|Comparison of several types of [[graphical projection]]]]
[[File:Various projections of cube above plane.svg|thumb|Various projections and how they are produced]]
[[Image:Axonometric projections.png|thumb|The three views. The percentages show the amount of foreshortening.]]

A simple orthographic [[projection (linear algebra)|projection]] onto the [[plane (mathematics)|plane]] ''z'' = 0 can be defined by the following matrix:
:<math>
P = 
\begin{bmatrix}
1 & 0 & 0  \\
0 & 1 & 0  \\
0 & 0 & 0  \\
\end{bmatrix}
</math>

For each point ''v'' = (''v''<sub>''x''</sub>, ''v''<sub>''y''</sub>, ''v''<sub>''z''</sub>), the transformed point ''Pv'' would be
:<math>
Pv = 
\begin{bmatrix}
1 & 0 & 0  \\
0 & 1 & 0  \\
0 & 0 & 0  \\
\end{bmatrix}
\begin{bmatrix}
v_x \\ v_y \\ v_z
\end{bmatrix}
=
\begin{bmatrix}
v_x \\ v_y \\ 0
\end{bmatrix}
</math>

Often, it is more useful to use [[homogeneous coordinates]].  The transformation above can be represented for homogeneous coordinates as 
:<math>
P = 
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
</math>

For each homogeneous vector ''v'' = (''v''<sub>''x''</sub>, ''v''<sub>''y''</sub>, ''v''<sub>''z''</sub>, 1), the transformed vector ''Pv'' would be
:<math>
Pv = 
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
v_x \\ v_y \\ v_z \\ 1
\end{bmatrix}
=
\begin{bmatrix}
v_x \\ v_y \\ 0 \\ 1
\end{bmatrix}
</math>

In [[computer graphics]],  one of the most common matrices used for orthographic [[projection (linear algebra)|projection]] can be defined by a [[n-tuple|6-tuple]], (''left'', ''right'', ''bottom'', ''top'', ''near'', ''far''), which defines the [[clipping (computer graphics)|clipping]] planes.  These planes form a box with the minimum corner at (''left'', ''bottom'', -''near'') and the maximum corner at (''right'', ''top'', -''far'').<ref>{{Cite web |last=Thormählen |first=Thorsten |date=November 26, 2021 |title=Graphics Programming – Cameras: Parallel Projection – Part 6, Chapter 2 |url=https://www.mathematik.uni-marburg.de/~thormae/lectures/graphics1/graphics_6_2_eng_web.html#10 |access-date=2022-04-22 |website=Mathematik Uni Marburg |pages=8 ff}}</ref>

The box is translated so that its center is at the origin, then it is scaled to the unit cube which is defined by having a minimum corner at (−1,−1,−1) and a maximum corner at (1,1,1).

The orthographic transform can be given by the following matrix:
:<math>
P = 
\begin{bmatrix}
\frac{2}{\text{right}-\text{left}} & 0 & 0 & -\frac{\text{right}+\text{left}}{\text{right}-\text{left}} \\
0 & \frac{2}{\text{top}-\text{bottom}} & 0 & -\frac{\text{top}+\text{bottom}}{\text{top}-\text{bottom}} \\
0 & 0 & \frac{-2}{\text{far}-\text{near}} & -\frac{\text{far}+\text{near}}{\text{far}-\text{near}} \\
0 & 0 & 0 & 1
\end{bmatrix}
</math>
which can be given as a [[scaling (geometry)|scaling]] ''S'' followed by a [[translation (geometry)|translation]] ''T'' of the form
:<math>
P = ST = 
\begin{bmatrix}
\frac{2}{\text{right}-\text{left}} & 0 & 0 & 0 \\
0 & \frac{2}{\text{top}-\text{bottom}} & 0 & 0 \\
0 & 0 & \frac{2}{\text{far}-\text{near}} & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
1 & 0 & 0 & -\frac{\text{left}+\text{right}}{2} \\
0 & 1 & 0 & -\frac{\text{top}+\text{bottom}}{2} \\
0 & 0 & -1 & -\frac{\text{far}+\text{near}}{2} \\
0 & 0 & 0 & 1
\end{bmatrix}
</math>

The inversion of the projection matrix ''P<sup>−1</sup>'', which can be used as the unprojection matrix is defined:

<math>
P^{-1} = 
\begin{bmatrix}
\frac{\text{right}-\text{left}}{2} & 0 & 0 & \frac{\text{left}+\text{right}}{2} \\
0 & \frac{\text{top}-\text{bottom}}{2} & 0 & \frac{\text{top}+\text{bottom}}{2} \\
0 & 0 & \frac{\text{far}-\text{near}}{-2} &  -\frac{\text{far}+\text{near}}{2} \\
0 & 0 & 0 & 1
\end{bmatrix}
</math>