Perspective Projection Derivation part.2

Perspective Part.2

There are hosts of references that use fov, aspect, far, near to discribe the Perspective Projection Matrix.

The four parameters represent respectively:

fov: field of view

aspect: aspect ratio

far: far plane

near: near plane

Here we use the vertical FOV, and transform those to previous parameters, t, b, l, r, n, f.

$$\begin{align*} &\left\{\begin{matrix} n = near\\ f = far \end{matrix}\right.\\\\ &\left\{\begin{matrix} t = n \times \tan{\frac{fov}{2}}\\ b = -n \times \tan{\frac{fov}{2}} \end{matrix}\right.\\\\ &\left\{\begin{matrix} r = aspect \times n \times \tan{\frac{fov}{2}}\\ l = -aspect \times n \times \tan{\frac{fov}{2}} \end{matrix}\right. \end{align*}$$

Eventually, the previous formula is transformed into:

$$\begin{pmatrix} \frac{\cot{\frac{for}{2}}}{aspect \cdot n} & 0 & 0 & 0\\ 0 & \frac{\cot{\frac{for}{2}}}{near} & 0 & 0\\ 0 & 0 & \frac{2}{near-far} & 0\\ 0 & 0 & 0 & 1 \end{pmatrix}\times \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & -\frac{near+far}{2}\\ 0 & 0 & 0 & 1 \end{pmatrix}\times \begin{pmatrix} near & 0 & 0 & 0\\ 0 & near & 0 & 0\\ 0 & 0 & near+far & -near*far\\ 0 & 0 & 1 & 0 \end{pmatrix}\\ =\begin{pmatrix} \frac{\cot{\frac{for}{2}}}{aspect} & 0 & 0 & 0\\ 0 & \cot{\frac{fov}{2}} & 0 & 0\\ 0 & 0 & \frac{near+far}{near-far} & -\frac{2\cdot near \cdot far}{near - far}\\ 0 & 0 & 1 & 0 \end{pmatrix}$$