## Orthographic projection

The orthographic projection is the simplest projection. It resembles architectural or design drawings in that it lacks depth scaling- farther objects are not smaller.

To perform the orthographic projection, the center of the view volume needs to be moved the origin. Then, the view volume must be scaled to fit into the OpenGL view volume: \( (-1, -1, -1) - (1, 1, 1) \).

If the bounds of the orthographic view volume are defined as \(left\), \(right\), \(bottom\), \(top\), \(near\), and \(var\), then the vector to translate to the origin is:

$$ \begin{bmatrix} -\frac{r+l}{2} & -\frac{t+b}{2} & -\frac{n+f}{2} & \end{bmatrix} ^T $$

Forming the translation matrix: $$ \begin{bmatrix} 1 & 0 & 0 & -\frac{r+l}{2} \ 0 & 1 & 0 & -\frac{t+b}{2} \ 0 & 0 & 1 & -\frac{n+f}{2} \ 0 & 0 & 0 & 1 \end{bmatrix} $$

The view volume must then be scaled by its width, height, and depth. $$ \begin{bmatrix} \frac{2}{r-l} & 0 & 0 & 0 \ 0 & \frac{2}{t-b} & 0 & 0 \ 0 & 0 & \frac{2}{n-f} & 0 \ 0 & 0 & 0 & 1 \end{bmatrix} $$

The final orthographic projection is: $$ \mathbf{O} = \begin{bmatrix} \frac{2}{r-l} & 0 & 0 & 0 \ 0 & \frac{2}{t-b} & 0 & 0 \ 0 & 0 & \frac{2}{n-f} & 0 \ 0 & 0 & 0 & 1 \end{bmatrix}

# \begin{bmatrix} 1 & 0 & 0 & -\frac{r+l}{2} \ 0 & 1 & 0 & -\frac{t+b}{2} \ 0 & 0 & 1 & -\frac{n+f}{2} \ 0 & 0 & 0 & 1 \end{bmatrix}

\begin{bmatrix} \frac{2}{r-l} & 0 & 0 & -\frac{r+l}{r-l} \ 0 & \frac{2}{t-b} & 0 & -\frac{t+b}{t-b} \ 0 & 0 & \frac{2}{n-f} & -\frac{n+f}{n-f} \ 0 & 0 & 0 & 1 \end{bmatrix} $$

### Properties

It is helpful to compare the orthographic projection to the perspective projection we observe with our eyes. In the perspective projection, parallel lines appear to converge at infinity; in the orthographic projection, parallel lines remain parallel. In perspective projection, the view volume is a frustum (a clipped pyramid with the eye where the top would be). In orthographic projection, the view volume is rectangular prism.