Pre-multiplication, right-handed coordinate system
Axis in matrix
[ X-Axis → m 11 m 12 m 13 m 14 Y-Axis → m 21 m 22 m 23 m 24 Z-Axis → m 31 m 32 m 33 m 34 Translate → m 41 m 42 m 43 m 44 ] \begin{array}{c} \begin{bmatrix*}[r] \color{red} \textrm{X-Axis} \rightarrow & \color{red} \mathbf{m_{11}} & \color{red} \mathbf{m_{12}} & \color{red} \mathbf{m_{13}} & \color{gray} m_{14}\\ \color{green} \textrm{Y-Axis} \rightarrow & \color{green} \mathbf{m_{21}} & \color{green} \mathbf{m_{22}} & \color{green} \mathbf{m_{23}} & \color{gray} m_{24}\\ \color{blue} \textrm{Z-Axis} \rightarrow & \color{blue} \mathbf{m_{31}} & \color{blue} \mathbf{m_{32}} & \color{blue} \mathbf{m_{33}} & \color{gray} m_{34}\\ \textrm{Translate} \rightarrow & \mathbf{m_{41}} & \mathbf{m_{42}} & \mathbf{m_{43}} & \color{gray} m_{44}\\ \end{bmatrix*} \end{array} ⎣ ⎡ X-Axis → Y-Axis → Z-Axis → Translate → m 11 m 21 m 31 m 41 m 12 m 22 m 32 m 42 m 13 m 23 m 33 m 43 m 14 m 24 m 34 m 44 ⎦ ⎤
l l l
r r r
b b b
t t t
n n n
f f f
w w w
h h h
α \alpha α
β \beta β
Perspective Transformation [ n , f ] \mathbf{\lbrack n,f \rbrack} [ n , f ] z ∈ [ 0 , 1 ] \mathbf{z \in \lbrack 0,1 \rbrack} z ∈ [ 0 , 1 ]
View Space
Left-handed NDC Space
Right-handed NDC Space
Frustum Symmetric Vertical FOV Symmetric Horizontal FOV [ 2 n r − l 0 0 0 0 2 n t − b 0 0 r + l r − l t + b t − b − f f − n − 1 0 0 − f n f − n 0 ] [ h w cot ( α 2 ) 0 0 0 0 cot ( α 2 ) 0 0 0 0 − f f − n − 1 0 0 − f n f − n 0 ] [ cot ( β 2 ) 0 0 0 0 w h cot ( β 2 ) 0 0 0 0 − f f − n − 1 0 0 − f n f − n 0 ] \begin{array}{ccc} \textrm{Frustum} & \textrm{Symmetric Vertical FOV} & \textrm{Symmetric Horizontal FOV} \\ \begin{bmatrix} \frac{2n}{r-l} & 0 & 0 & 0\\ 0 & \frac{2n}{t-b} & 0 & 0\\ \frac{r+l}{r-l} & \frac{t+b}{t-b} & -\frac{f}{f-n} & -1\\ 0 & 0 & -\frac{fn}{f-n} & 0\\ \end{bmatrix} & \begin{bmatrix} \frac{h}{w}\cot(\frac{\alpha}{2}) & 0 & 0 & 0\\ 0 & \cot(\frac{\alpha}{2}) & 0 & 0\\ 0 & 0 & -\frac{f}{f-n} & -1\\ 0 & 0 & -\frac{fn}{f-n} & 0\\ \end{bmatrix} & \begin{bmatrix} \cot(\frac{\beta}{2}) & 0 & 0 & 0\\ 0 & \frac{w}{h}\cot(\frac{\beta}{2}) & 0 & 0\\ 0 & 0 & -\frac{f}{f-n} & -1\\ 0 & 0 & -\frac{fn}{f-n} & 0\\ \end{bmatrix} \end{array} Frustum ⎣ ⎡ r − l 2 n 0 r − l r + l 0 0 t − b 2 n t − b t + b 0 0 0 − f − n f − f − n f n 0 0 − 1 0 ⎦ ⎤ Symmetric Vertical FOV ⎣ ⎡ w h cot ( 2 α ) 0 0 0 0 cot ( 2 α ) 0 0 0 0 − f − n f − f − n f n 0 0 − 1 0 ⎦ ⎤ Symmetric Horizontal FOV ⎣ ⎡ cot ( 2 β ) 0 0 0 0 h w cot ( 2 β ) 0 0 0 0 − f − n f − f − n f n 0 0 − 1 0 ⎦ ⎤
Perspective Transformation infinite far plane [ n , ∞ ] \mathbf{\lbrack n,\infty \rbrack} [ n , ∞ ] z ∈ [ 0 , 1 ] \mathbf{z \in \lbrack 0,1 \rbrack} z ∈ [ 0 , 1 ]
View Space
Left-handed NDC Space
Right-handed NDC Space
Frustum Symmetric Vertical FOV Symmetric Horizontal FOV [ 2 n r − l 0 0 0 0 2 n t − b 0 0 r + l r − l t + b t − b − 1 − 1 0 0 − n 0 ] [ h w cot ( α 2 ) 0 0 0 0 cot ( α 2 ) 0 0 0 0 − 1 − 1 0 0 − n 0 ] [ cot ( β 2 ) 0 0 0 0 w h cot ( β 2 ) 0 0 0 0 − 1 − 1 0 0 − n 0 ] \begin{array}{ccc} \textrm{Frustum} & \textrm{Symmetric Vertical FOV} & \textrm{Symmetric Horizontal FOV} \\ \begin{bmatrix} \frac{2n}{r-l} & 0 & 0 & 0\\ 0 & \frac{2n}{t-b} & 0 & 0\\ \frac{r+l}{r-l} & \frac{t+b}{t-b} & -1 & -1\\ 0 & 0 & -n & 0\\ \end{bmatrix} & \begin{bmatrix} \frac{h}{w}\cot(\frac{\alpha}{2}) & 0 & 0 & 0\\ 0 & \cot(\frac{\alpha}{2}) & 0 & 0\\ 0 & 0 & -1 & -1\\ 0 & 0 & -n & 0\\ \end{bmatrix} & \begin{bmatrix} \cot(\frac{\beta}{2}) & 0 & 0 & 0\\ 0 & \frac{w}{h}\cot(\frac{\beta}{2}) & 0 & 0\\ 0 & 0 & -1 & -1\\ 0 & 0 & -n & 0\\ \end{bmatrix} \end{array} Frustum ⎣ ⎡ r − l 2 n 0 r − l r + l 0 0 t − b 2 n t − b t + b 0 0 0 − 1 − n 0 0 − 1 0 ⎦ ⎤ Symmetric Vertical FOV ⎣ ⎡ w h cot ( 2 α ) 0 0 0 0 cot ( 2 α ) 0 0 0 0 − 1 − n 0 0 − 1 0 ⎦ ⎤ Symmetric Horizontal FOV ⎣ ⎡ cot ( 2 β ) 0 0 0 0 h w cot ( 2 β ) 0 0 0 0 − 1 − n 0 0 − 1 0 ⎦ ⎤
Perspective Transformation [ n , f ] \mathbf{\lbrack n,f \rbrack} [ n , f ] z ∈ [ 1 , 0 ] \mathbf{z \in \lbrack 1,0 \rbrack} z ∈ [ 1 , 0 ]
View Space
Left-handed NDC Space
Right-handed NDC Space
Frustum Symmetric Vertical FOV Symmetric Horizontal FOV [ 2 n r − l 0 0 0 0 2 n t − b 0 0 r + l r − l t + b t − b n f − n − 1 0 0 f n f − n 0 ] [ h w cot ( α 2 ) 0 0 0 0 cot ( α 2 ) 0 0 0 0 n f − n − 1 0 0 f n f − n 0 ] [ cot ( β 2 ) 0 0 0 0 w h cot ( β 2 ) 0 0 0 0 n f − n − 1 0 0 f n f − n 0 ] \begin{array}{ccc} \textrm{Frustum} & \textrm{Symmetric Vertical FOV} & \textrm{Symmetric Horizontal FOV} \\ \begin{bmatrix} \frac{2n}{r-l} & 0 & 0 & 0\\ 0 & \frac{2n}{t-b} & 0 & 0\\ \frac{r+l}{r-l} & \frac{t+b}{t-b} & \frac{n}{f-n} & -1\\ 0 & 0 & \frac{fn}{f-n} & 0\\ \end{bmatrix} & \begin{bmatrix} \frac{h}{w}\cot(\frac{\alpha}{2}) & 0 & 0 & 0\\ 0 & \cot(\frac{\alpha}{2}) & 0 & 0\\ 0 & 0 & \frac{n}{f-n} & -1\\ 0 & 0 & \frac{fn}{f-n} & 0\\ \end{bmatrix} & \begin{bmatrix} \cot(\frac{\beta}{2}) & 0 & 0 & 0\\ 0 & \frac{w}{h}\cot(\frac{\beta}{2}) & 0 & 0\\ 0 & 0 & \frac{n}{f-n} & -1\\ 0 & 0 & \frac{fn}{f-n} & 0\\ \end{bmatrix} \end{array} Frustum ⎣ ⎡ r − l 2 n 0 r − l r + l 0 0 t − b 2 n t − b t + b 0 0 0 f − n n f − n f n 0 0 − 1 0 ⎦ ⎤ Symmetric Vertical FOV ⎣ ⎡ w h cot ( 2 α ) 0 0 0 0 cot ( 2 α ) 0 0 0 0 f − n n f − n f n 0 0 − 1 0 ⎦ ⎤ Symmetric Horizontal FOV ⎣ ⎡ cot ( 2 β ) 0 0 0 0 h w cot ( 2 β ) 0 0 0 0 f − n n f − n f n 0 0 − 1 0 ⎦ ⎤
Perspective Transformation infinite far plane [ n , ∞ ] \mathbf{\lbrack n,\infty \rbrack} [ n , ∞ ] z ∈ [ 1 , 0 ] \mathbf{z \in \lbrack 1,0 \rbrack} z ∈ [ 1 , 0 ]
View Space
Left-handed NDC Space
Right-handed NDC Space
Frustum Symmetric Vertical FOV Symmetric Horizontal FOV [ 2 n r − l 0 0 0 0 2 n t − b 0 0 r + l r − l t + b t − b 0 − 1 0 0 n 0 ] [ h w cot ( α 2 ) 0 0 0 0 cot ( α 2 ) 0 0 0 0 0 − 1 0 0 n 0 ] [ cot ( β 2 ) 0 0 0 0 w h cot ( β 2 ) 0 0 0 0 0 − 1 0 0 n 0 ] \begin{array}{ccc} \textrm{Frustum} & \textrm{Symmetric Vertical FOV} & \textrm{Symmetric Horizontal FOV} \\ \begin{bmatrix} \frac{2n}{r-l} & 0 & 0 & 0\\ 0 & \frac{2n}{t-b} & 0 & 0\\ \frac{r+l}{r-l} & \frac{t+b}{t-b} & 0 & -1\\ 0 & 0 & n & 0\\ \end{bmatrix} & \begin{bmatrix} \frac{h}{w}\cot(\frac{\alpha}{2}) & 0 & 0 & 0\\ 0 & \cot(\frac{\alpha}{2}) & 0 & 0\\ 0 & 0 & 0 & -1\\ 0 & 0 & n & 0\\ \end{bmatrix} & \begin{bmatrix} \cot(\frac{\beta}{2}) & 0 & 0 & 0\\ 0 & \frac{w}{h}\cot(\frac{\beta}{2}) & 0 & 0\\ 0 & 0 & 0 & -1\\ 0 & 0 & n & 0\\ \end{bmatrix} \end{array} Frustum ⎣ ⎡ r − l 2 n 0 r − l r + l 0 0 t − b 2 n t − b t + b 0 0 0 0 n 0 0 − 1 0 ⎦ ⎤ Symmetric Vertical FOV ⎣ ⎡ w h cot ( 2 α ) 0 0 0 0 cot ( 2 α ) 0 0 0 0 0 n 0 0 − 1 0 ⎦ ⎤ Symmetric Horizontal FOV ⎣ ⎡ cot ( 2 β ) 0 0 0 0 h w cot ( 2 β ) 0 0 0 0 0 n 0 0 − 1 0 ⎦ ⎤
Perspective Transformation [ n , f ] \mathbf{\lbrack n,f \rbrack} [ n , f ] z ∈ [ − 1 , 1 ] \mathbf{z \in \lbrack -1,1 \rbrack} z ∈ [ − 1 , 1 ]
View Space
Left-handed NDC Space
Right-handed NDC Space
Frustum Symmetric Vertical FOV Symmetric Horizontal FOV [ 2 n r − l 0 0 0 0 2 n t − b 0 0 r + l r − l t + b t − b − f + n f − n − 1 0 0 − 2 f n f − n 0 ] [ h w cot ( α 2 ) 0 0 0 0 cot ( α 2 ) 0 0 0 0 − f + n f − n − 1 0 0 − 2 f n f − n 0 ] [ cot ( β 2 ) 0 0 0 0 w h cot ( β 2 ) 0 0 0 0 − f + n f − n − 1 0 0 − 2 f n f − n 0 ] \begin{array}{ccc} \textrm{Frustum} & \textrm{Symmetric Vertical FOV} & \textrm{Symmetric Horizontal FOV} \\ \begin{bmatrix} \frac{2n}{r-l} & 0 & 0 & 0\\ 0 & \frac{2n}{t-b} & 0 & 0\\ \frac{r+l}{r-l} & \frac{t+b}{t-b} & -\frac{f+n}{f-n} & -1\\ 0 & 0 & -\frac{2fn}{f-n} & 0\\ \end{bmatrix} & \begin{bmatrix} \frac{h}{w}\cot(\frac{\alpha}{2}) & 0 & 0 & 0\\ 0 & \cot(\frac{\alpha}{2}) & 0 & 0\\ 0 & 0 & -\frac{f+n}{f-n} & -1\\ 0 & 0 & -\frac{2fn}{f-n} & 0\\ \end{bmatrix} & \begin{bmatrix} \cot(\frac{\beta}{2}) & 0 & 0 & 0\\ 0 & \frac{w}{h}\cot(\frac{\beta}{2}) & 0 & 0\\ 0 & 0 & -\frac{f+n}{f-n} & -1\\ 0 & 0 & -\frac{2fn}{f-n} & 0\\ \end{bmatrix} \end{array} Frustum ⎣ ⎡ r − l 2 n 0 r − l r + l 0 0 t − b 2 n t − b t + b 0 0 0 − f − n f + n − f − n 2 f n 0 0 − 1 0 ⎦ ⎤ Symmetric Vertical FOV ⎣ ⎡ w h cot ( 2 α ) 0 0 0 0 cot ( 2 α ) 0 0 0 0 − f − n f + n − f − n 2 f n 0 0 − 1 0 ⎦ ⎤ Symmetric Horizontal FOV ⎣ ⎡ cot ( 2 β ) 0 0 0 0 h w cot ( 2 β ) 0 0 0 0 − f − n f + n − f − n 2 f n 0 0 − 1 0 ⎦ ⎤
Perspective Transformation infinite far plane [ n , ∞ ] \mathbf{\lbrack n,\infty \rbrack} [ n , ∞ ] z ∈ [ − 1 , 1 ] \mathbf{z \in \lbrack -1,1 \rbrack} z ∈ [ − 1 , 1 ]
View Space
Left-handed NDC Space
Right-handed NDC Space
Frustum Symmetric Vertical FOV Symmetric Horizontal FOV [ 2 n r − l 0 0 0 0 2 n t − b 0 0 r + l r − l t + b t − b − 1 − 1 0 0 − 2 n 0 ] [ h w cot ( α 2 ) 0 0 0 0 cot ( α 2 ) 0 0 0 0 − 1 − 1 0 0 − 2 n 0 ] [ cot ( β 2 ) 0 0 0 0 w h cot ( β 2 ) 0 0 0 0 − 1 − 1 0 0 − 2 n 0 ] \begin{array}{ccc} \textrm{Frustum} & \textrm{Symmetric Vertical FOV} & \textrm{Symmetric Horizontal FOV} \\ \begin{bmatrix} \frac{2n}{r-l} & 0 & 0 & 0\\ 0 & \frac{2n}{t-b} & 0 & 0\\ \frac{r+l}{r-l} & \frac{t+b}{t-b} & -1 & -1\\ 0 & 0 & -2n & 0\\ \end{bmatrix} & \begin{bmatrix} \frac{h}{w}\cot(\frac{\alpha}{2}) & 0 & 0 & 0\\ 0 & \cot(\frac{\alpha}{2}) & 0 & 0\\ 0 & 0 & -1 & -1\\ 0 & 0 & -2n & 0\\ \end{bmatrix} & \begin{bmatrix} \cot(\frac{\beta}{2}) & 0 & 0 & 0\\ 0 & \frac{w}{h}\cot(\frac{\beta}{2}) & 0 & 0\\ 0 & 0 & -1 & -1\\ 0 & 0 & -2n & 0\\ \end{bmatrix} \end{array} Frustum ⎣ ⎡ r − l 2 n 0 r − l r + l 0 0 t − b 2 n t − b t + b 0 0 0 − 1 − 2 n 0 0 − 1 0 ⎦ ⎤ Symmetric Vertical FOV ⎣ ⎡ w h cot ( 2 α ) 0 0 0 0 cot ( 2 α ) 0 0 0 0 − 1 − 2 n 0 0 − 1 0 ⎦ ⎤ Symmetric Horizontal FOV ⎣ ⎡ cot ( 2 β ) 0 0 0 0 h w cot ( 2 β ) 0 0 0 0 − 1 − 2 n 0 0 − 1 0 ⎦ ⎤
Perspective Transformation [ n , f ] \mathbf{\lbrack n,f \rbrack} [ n , f ] z ∈ [ 1 , − 1 ] \mathbf{z \in \lbrack 1,-1 \rbrack} z ∈ [ 1 , − 1 ]
View Space
Left-handed NDC Space
Right-handed NDC Space
Frustum Symmetric Vertical FOV Symmetric Horizontal FOV [ 2 n r − l 0 0 0 0 2 n t − b 0 0 r + l r − l t + b t − b f + n f − n − 1 0 0 2 f n f − n 0 ] [ h w cot ( α 2 ) 0 0 0 0 cot ( α 2 ) 0 0 0 0 f + n f − n − 1 0 0 2 f n f − n 0 ] [ cot ( β 2 ) 0 0 0 0 w h cot ( β 2 ) 0 0 0 0 f + n f − n − 1 0 0 2 f n f − n 0 ] \begin{array}{ccc} \textrm{Frustum} & \textrm{Symmetric Vertical FOV} & \textrm{Symmetric Horizontal FOV} \\ \begin{bmatrix} \frac{2n}{r-l} & 0 & 0 & 0\\ 0 & \frac{2n}{t-b} & 0 & 0\\ \frac{r+l}{r-l} & \frac{t+b}{t-b} & \frac{f+n}{f-n} & -1\\ 0 & 0 & \frac{2fn}{f-n} & 0\\ \end{bmatrix} & \begin{bmatrix} \frac{h}{w}\cot(\frac{\alpha}{2}) & 0 & 0 & 0\\ 0 & \cot(\frac{\alpha}{2}) & 0 & 0\\ 0 & 0 & \frac{f+n}{f-n} & -1\\ 0 & 0 & \frac{2fn}{f-n} & 0\\ \end{bmatrix} & \begin{bmatrix} \cot(\frac{\beta}{2}) & 0 & 0 & 0\\ 0 & \frac{w}{h}\cot(\frac{\beta}{2}) & 0 & 0\\ 0 & 0 & \frac{f+n}{f-n} & -1\\ 0 & 0 & \frac{2fn}{f-n} & 0\\ \end{bmatrix} \end{array} Frustum ⎣ ⎡ r − l 2 n 0 r − l r + l 0 0 t − b 2 n t − b t + b 0 0 0 f − n f + n f − n 2 f n 0 0 − 1 0 ⎦ ⎤ Symmetric Vertical FOV ⎣ ⎡ w h cot ( 2 α ) 0 0 0 0 cot ( 2 α ) 0 0 0 0 f − n f + n f − n 2 f n 0 0 − 1 0 ⎦ ⎤ Symmetric Horizontal FOV ⎣ ⎡ cot ( 2 β ) 0 0 0 0 h w cot ( 2 β ) 0 0 0 0 f − n f + n f − n 2 f n 0 0 − 1 0 ⎦ ⎤
Perspective Transformation infinite far plane [ n , ∞ ] \mathbf{\lbrack n,\infty \rbrack} [ n , ∞ ] z ∈ [ 1 , − 1 ] \mathbf{z \in \lbrack 1,-1 \rbrack} z ∈ [ 1 , − 1 ]
View Space
Left-handed NDC Space
Right-handed NDC Space
Frustum Symmetric Vertical FOV Symmetric Horizontal FOV [ 2 n r − l 0 0 0 0 2 n t − b 0 0 r + l r − l t + b t − b 1 2 n 0 0 − 1 0 ] [ h w cot ( α 2 ) 0 0 0 0 cot ( α 2 ) 0 0 0 0 1 − 1 0 0 2 n 0 ] [ cot ( β 2 ) 0 0 0 0 w h cot ( β 2 ) 0 0 0 0 1 − 1 0 0 2 n 0 ] \begin{array}{ccc} \textrm{Frustum} & \textrm{Symmetric Vertical FOV} & \textrm{Symmetric Horizontal FOV} \\ \begin{bmatrix} \frac{2n}{r-l} & 0 & 0 & 0\\ 0 & \frac{2n}{t-b} & 0 & 0\\ \frac{r+l}{r-l} & \frac{t+b}{t-b} & 1 & 2n\\ 0 & 0 & -1 & 0\\ \end{bmatrix} & \begin{bmatrix} \frac{h}{w}\cot(\frac{\alpha}{2}) & 0 & 0 & 0\\ 0 & \cot(\frac{\alpha}{2}) & 0 & 0\\ 0 & 0 & 1 & -1\\ 0 & 0 & 2n & 0\\ \end{bmatrix} & \begin{bmatrix} \cot(\frac{\beta}{2}) & 0 & 0 & 0\\ 0 & \frac{w}{h}\cot(\frac{\beta}{2}) & 0 & 0\\ 0 & 0 & 1 & -1\\ 0 & 0 & 2n & 0\\ \end{bmatrix} \end{array} Frustum ⎣ ⎡ r − l 2 n 0 r − l r + l 0 0 t − b 2 n t − b t + b 0 0 0 1 − 1 0 0 2 n 0 ⎦ ⎤ Symmetric Vertical FOV ⎣ ⎡ w h cot ( 2 α ) 0 0 0 0 cot ( 2 α ) 0 0 0 0 1 2 n 0 0 − 1 0 ⎦ ⎤ Symmetric Horizontal FOV ⎣ ⎡ cot ( 2 β ) 0 0 0 0 h w cot ( 2 β ) 0 0 0 0 1 2 n 0 0 − 1 0 ⎦ ⎤
l l l
r r r
b b b
t t t
n n n
f f f
w w w
h h h
j x j_x j x
j y j_y j y
α \alpha α
β \beta β
Jitter Perspective Transformation [ n , f ] \mathbf{\lbrack n,f \rbrack} [ n , f ] z ∈ [ 0 , 1 ] \mathbf{z \in \lbrack 0,1 \rbrack} z ∈ [ 0 , 1 ]
Frustum Symmetric Vertical FOV Symmetric Horizontal FOV [ 2 n r − l 0 0 0 0 2 n t − b 0 0 r ( 1 + j x ) + l ( 1 − j x ) r − l t ( 1 + j y ) + b ( 1 − j y ) t − b − f f − n − 1 0 0 − f n f − n 0 ] [ h w cot ( α 2 ) 0 0 0 0 cot ( α 2 ) 0 0 j x j y − f f − n − 1 0 0 − f n f − n 0 ] [ cot ( β 2 ) 0 0 0 0 w h cot ( β 2 ) 0 0 j x j y − f f − n − 1 0 0 − f n f − n 0 ] \begin{array}{ccc} \textrm{Frustum} & \textrm{Symmetric Vertical FOV} & \textrm{Symmetric Horizontal FOV} \\ \begin{bmatrix} \frac{2n}{r-l} & 0 & 0 & 0\\ 0 & \frac{2n}{t-b} & 0 & 0\\ \frac{r(1+j_x)+l(1-j_x)}{r-l} & \frac{t(1+j_y)+b(1-j_y)}{t-b} & -\frac{f}{f-n} & -1\\ 0 & 0 & -\frac{fn}{f-n} & 0\\ \end{bmatrix} & \begin{bmatrix} \frac{h}{w}\cot(\frac{\alpha}{2}) & 0 & 0 & 0\\ 0 & \cot(\frac{\alpha}{2}) & 0 & 0\\ j_x & j_y & -\frac{f}{f-n} & -1\\ 0 & 0 & -\frac{fn}{f-n} & 0\\ \end{bmatrix} & \begin{bmatrix} \cot(\frac{\beta}{2}) & 0 & 0 & 0\\ 0 & \frac{w}{h}\cot(\frac{\beta}{2}) & 0 & 0\\ j_x & j_y & -\frac{f}{f-n} & -1\\ 0 & 0 & -\frac{fn}{f-n} & 0\\ \end{bmatrix} \end{array} Frustum ⎣ ⎡ r − l 2 n 0 r − l r ( 1 + j x ) + l ( 1 − j x ) 0 0 t − b 2 n t − b t ( 1 + j y ) + b ( 1 − j y ) 0 0 0 − f − n f − f − n f n 0 0 − 1 0 ⎦ ⎤ Symmetric Vertical FOV ⎣ ⎡ w h cot ( 2 α ) 0 j x 0 0 cot ( 2 α ) j y 0 0 0 − f − n f − f − n f n 0 0 − 1 0 ⎦ ⎤ Symmetric Horizontal FOV ⎣ ⎡ cot ( 2 β ) 0 j x 0 0 h w cot ( 2 β ) j y 0 0 0 − f − n f − f − n f n 0 0 − 1 0 ⎦ ⎤
Jitter Perspective Transformation infinite far plane [ n , ∞ ] \mathbf{\lbrack n,\infty \rbrack} [ n , ∞ ] z ∈ [ 0 , 1 ] \mathbf{z \in \lbrack 0,1 \rbrack} z ∈ [ 0 , 1 ]
Frustum Symmetric Vertical FOV Symmetric Horizontal FOV [ 2 n r − l 0 0 0 0 2 n t − b 0 0 r ( 1 + j x ) + l ( 1 − j x ) r − l t ( 1 + j y ) + b ( 1 − j y ) t − b − 1 − 1 0 0 − n 0 ] [ h w cot ( α 2 ) 0 0 0 0 cot ( α 2 ) 0 0 j x j y − 1 − 1 0 0 − n 0 ] [ cot ( β 2 ) 0 0 0 0 w h cot ( β 2 ) 0 0 j x j y − 1 − 1 0 0 − n 0 ] \begin{array}{ccc} \textrm{Frustum} & \textrm{Symmetric Vertical FOV} & \textrm{Symmetric Horizontal FOV} \\ \begin{bmatrix} \frac{2n}{r-l} & 0 & 0 & 0\\ 0 & \frac{2n}{t-b} & 0 & 0\\ \frac{r(1+j_x)+l(1-j_x)}{r-l} & \frac{t(1+j_y)+b(1-j_y)}{t-b} & -1 & -1\\ 0 & 0 & -n & 0\\ \end{bmatrix} & \begin{bmatrix} \frac{h}{w}\cot(\frac{\alpha}{2}) & 0 & 0 & 0\\ 0 & \cot(\frac{\alpha}{2}) & 0 & 0\\ j_x & j_y & -1 & -1\\ 0 & 0 & -n & 0\\ \end{bmatrix} & \begin{bmatrix} \cot(\frac{\beta}{2}) & 0 & 0 & 0\\ 0 & \frac{w}{h}\cot(\frac{\beta}{2}) & 0 & 0\\ j_x & j_y & -1 & -1\\ 0 & 0 & -n & 0\\ \end{bmatrix} \end{array} Frustum ⎣ ⎡ r − l 2 n 0 r − l r ( 1 + j x ) + l ( 1 − j x ) 0 0 t − b 2 n t − b t ( 1 + j y ) + b ( 1 − j y ) 0 0 0 − 1 − n 0 0 − 1 0 ⎦ ⎤ Symmetric Vertical FOV ⎣ ⎡ w h cot ( 2 α ) 0 j x 0 0 cot ( 2 α ) j y 0 0 0 − 1 − n 0 0 − 1 0 ⎦ ⎤ Symmetric Horizontal FOV ⎣ ⎡ cot ( 2 β ) 0 j x 0 0 h w cot ( 2 β ) j y 0 0 0 − 1 − n 0 0 − 1 0 ⎦ ⎤
Jitter Perspective Transformation [ n , f ] \mathbf{\lbrack n,f \rbrack} [ n , f ] z ∈ [ 1 , 0 ] \mathbf{z \in \lbrack 1,0 \rbrack} z ∈ [ 1 , 0 ]
Frustum Symmetric Vertical FOV Symmetric Horizontal FOV [ 2 n r − l 0 0 0 0 2 n t − b 0 0 r ( 1 + j x ) + l ( 1 − j x ) r − l t ( 1 + j y ) + b ( 1 − j y ) t − b n f − n − 1 0 0 f n f − n 0 ] [ h w cot ( α 2 ) 0 0 0 0 cot ( α 2 ) 0 0 j x j y n f − n − 1 0 0 f n f − n 0 ] [ cot ( β 2 ) 0 0 0 0 w h cot ( β 2 ) 0 0 j x j y n f − n − 1 0 0 f n f − n 0 ] \begin{array}{ccc} \textrm{Frustum} & \textrm{Symmetric Vertical FOV} & \textrm{Symmetric Horizontal FOV} \\ \begin{bmatrix} \frac{2n}{r-l} & 0 & 0 & 0\\ 0 & \frac{2n}{t-b} & 0 & 0\\ \frac{r(1+j_x)+l(1-j_x)}{r-l} & \frac{t(1+j_y)+b(1-j_y)}{t-b} & \frac{n}{f-n} & -1\\ 0 & 0 & \frac{fn}{f-n} & 0\\ \end{bmatrix} & \begin{bmatrix} \frac{h}{w}\cot(\frac{\alpha}{2}) & 0 & 0 & 0\\ 0 & \cot(\frac{\alpha}{2}) & 0 & 0\\ j_x & j_y & \frac{n}{f-n} & -1\\ 0 & 0 & \frac{fn}{f-n} & 0\\ \end{bmatrix} & \begin{bmatrix} \cot(\frac{\beta}{2}) & 0 & 0 & 0\\ 0 & \frac{w}{h}\cot(\frac{\beta}{2}) & 0 & 0\\ j_x & j_y & \frac{n}{f-n} & -1\\ 0 & 0 & \frac{fn}{f-n} & 0\\ \end{bmatrix} \end{array} Frustum ⎣ ⎡ r − l 2 n 0 r − l r ( 1 + j x ) + l ( 1 − j x ) 0 0 t − b 2 n t − b t ( 1 + j y ) + b ( 1 − j y ) 0 0 0 f − n n f − n f n 0 0 − 1 0 ⎦ ⎤ Symmetric Vertical FOV ⎣ ⎡ w h cot ( 2 α ) 0 j x 0 0 cot ( 2 α ) j y 0 0 0 f − n n f − n f n 0 0 − 1 0 ⎦ ⎤ Symmetric Horizontal FOV ⎣ ⎡ cot ( 2 β ) 0 j x 0 0 h w cot ( 2 β ) j y 0 0 0 f − n n f − n f n 0 0 − 1 0 ⎦ ⎤
Jitter Perspective Transformation infinite far plane [ n , ∞ ] \mathbf{\lbrack n,\infty \rbrack} [ n , ∞ ] z ∈ [ 1 , 0 ] \mathbf{z \in \lbrack 1,0 \rbrack} z ∈ [ 1 , 0 ]
Frustum Symmetric Vertical FOV Symmetric Horizontal FOV [ 2 n r − l 0 0 0 0 2 n t − b 0 0 r ( 1 + j x ) + l ( 1 − j x ) r − l t ( 1 + j y ) + b ( 1 − j y ) t − b 0 − 1 0 0 n 0 ] [ h w cot ( α 2 ) 0 0 0 0 cot ( α 2 ) 0 0 j x j y 0 − 1 0 0 n 0 ] [ cot ( β 2 ) 0 0 0 0 w h cot ( β 2 ) 0 0 j x j y 0 − 1 0 0 n 0 ] \begin{array}{ccc} \textrm{Frustum} & \textrm{Symmetric Vertical FOV} & \textrm{Symmetric Horizontal FOV} \\ \begin{bmatrix} \frac{2n}{r-l} & 0 & 0 & 0\\ 0 & \frac{2n}{t-b} & 0 & 0\\ \frac{r(1+j_x)+l(1-j_x)}{r-l} & \frac{t(1+j_y)+b(1-j_y)}{t-b} & 0 & -1\\ 0 & 0 & n & 0\\ \end{bmatrix} & \begin{bmatrix} \frac{h}{w}\cot(\frac{\alpha}{2}) & 0 & 0 & 0\\ 0 & \cot(\frac{\alpha}{2}) & 0 & 0\\ j_x & j_y & 0 & -1\\ 0 & 0 & n & 0\\ \end{bmatrix} & \begin{bmatrix} \cot(\frac{\beta}{2}) & 0 & 0 & 0\\ 0 & \frac{w}{h}\cot(\frac{\beta}{2}) & 0 & 0\\ j_x & j_y & 0 & -1\\ 0 & 0 & n & 0\\ \end{bmatrix} \end{array} Frustum ⎣ ⎡ r − l 2 n 0 r − l r ( 1 + j x ) + l ( 1 − j x ) 0 0 t − b 2 n t − b t ( 1 + j y ) + b ( 1 − j y ) 0 0 0 0 n 0 0 − 1 0 ⎦ ⎤ Symmetric Vertical FOV ⎣ ⎡ w h cot ( 2 α ) 0 j x 0 0 cot ( 2 α ) j y 0 0 0 0 n 0 0 − 1 0 ⎦ ⎤ Symmetric Horizontal FOV ⎣ ⎡ cot ( 2 β ) 0 j x 0 0 h w cot ( 2 β ) j y 0 0 0 0 n 0 0 − 1 0 ⎦ ⎤
Jitter Perspective Transformation [ n , f ] \mathbf{\lbrack n,f \rbrack} [ n , f ] z ∈ [ − 1 , 1 ] \mathbf{z \in \lbrack -1,1 \rbrack} z ∈ [ − 1 , 1 ]
Frustum Symmetric Vertical FOV Symmetric Horizontal FOV [ 2 n r − l 0 0 0 0 2 n t − b 0 0 r ( 1 + j x ) + l ( 1 − j x ) r − l t ( 1 + j y ) + b ( 1 − j y ) t − b − f + n f − n − 1 0 0 − 2 f n f − n 0 ] [ h w cot ( α 2 ) 0 0 0 0 cot ( α 2 ) 0 0 j x j y − f + n f − n − 1 0 0 − 2 f n f − n 0 ] [ cot ( β 2 ) 0 0 0 0 w h cot ( β 2 ) 0 0 j x j y − f + n f − n − 1 0 0 − 2 f n f − n 0 ] \begin{array}{ccc} \textrm{Frustum} & \textrm{Symmetric Vertical FOV} & \textrm{Symmetric Horizontal FOV} \\ \begin{bmatrix} \frac{2n}{r-l} & 0 & 0 & 0\\ 0 & \frac{2n}{t-b} & 0 & 0\\ \frac{r(1+j_x)+l(1-j_x)}{r-l} & \frac{t(1+j_y)+b(1-j_y)}{t-b} & -\frac{f+n}{f-n} & -1\\ 0 & 0 & -\frac{2fn}{f-n} & 0\\ \end{bmatrix} & \begin{bmatrix} \frac{h}{w}\cot(\frac{\alpha}{2}) & 0 & 0 & 0\\ 0 & \cot(\frac{\alpha}{2}) & 0 & 0\\ j_x & j_y & -\frac{f+n}{f-n} & -1\\ 0 & 0 & -\frac{2fn}{f-n} & 0\\ \end{bmatrix} & \begin{bmatrix} \cot(\frac{\beta}{2}) & 0 & 0 & 0\\ 0 & \frac{w}{h}\cot(\frac{\beta}{2}) & 0 & 0\\ j_x & j_y & -\frac{f+n}{f-n} & -1\\ 0 & 0 & -\frac{2fn}{f-n} & 0\\ \end{bmatrix} \end{array} Frustum ⎣ ⎡ r − l 2 n 0 r − l r ( 1 + j x ) + l ( 1 − j x ) 0 0 t − b 2 n t − b t ( 1 + j y ) + b ( 1 − j y ) 0 0 0 − f − n f + n − f − n 2 f n 0 0 − 1 0 ⎦ ⎤ Symmetric Vertical FOV ⎣ ⎡ w h cot ( 2 α ) 0 j x 0 0 cot ( 2 α ) j y 0 0 0 − f − n f + n − f − n 2 f n 0 0 − 1 0 ⎦ ⎤ Symmetric Horizontal FOV ⎣ ⎡ cot ( 2 β ) 0 j x 0 0 h w cot ( 2 β ) j y 0 0 0 − f − n f + n − f − n 2 f n 0 0 − 1 0 ⎦ ⎤
Jitter Perspective Transformation infinite far plane [ n , ∞ ] \mathbf{\lbrack n,\infty \rbrack} [ n , ∞ ] z ∈ [ − 1 , 1 ] \mathbf{z \in \lbrack -1,1 \rbrack} z ∈ [ − 1 , 1 ]
Frustum Symmetric Vertical FOV Symmetric Horizontal FOV [ 2 n r − l 0 0 0 0 2 n t − b 0 0 r ( 1 + j x ) + l ( 1 − j x ) r − l t ( 1 + j y ) + b ( 1 − j y ) t − b − 1 − 1 0 0 − 2 n 0 ] [ h w cot ( α 2 ) 0 0 0 0 cot ( α 2 ) 0 0 j x j y − 1 − 1 0 0 − 2 n 0 ] [ cot ( β 2 ) 0 0 0 0 w h cot ( β 2 ) 0 0 j x j y − 1 − 1 0 0 − 2 n 0 ] \begin{array}{ccc} \textrm{Frustum} & \textrm{Symmetric Vertical FOV} & \textrm{Symmetric Horizontal FOV} \\ \begin{bmatrix} \frac{2n}{r-l} & 0 & 0 & 0\\ 0 & \frac{2n}{t-b} & 0 & 0\\ \frac{r(1+j_x)+l(1-j_x)}{r-l} & \frac{t(1+j_y)+b(1-j_y)}{t-b} & -1 & -1\\ 0 & 0 & -2n & 0\\ \end{bmatrix} & \begin{bmatrix} \frac{h}{w}\cot(\frac{\alpha}{2}) & 0 & 0 & 0\\ 0 & \cot(\frac{\alpha}{2}) & 0 & 0\\ j_x & j_y & -1 & -1\\ 0 & 0 & -2n & 0\\ \end{bmatrix} & \begin{bmatrix} \cot(\frac{\beta}{2}) & 0 & 0 & 0\\ 0 & \frac{w}{h}\cot(\frac{\beta}{2}) & 0 & 0\\ j_x & j_y & -1 & -1\\ 0 & 0 & -2n & 0\\ \end{bmatrix} \end{array} Frustum ⎣ ⎡ r − l 2 n 0 r − l r ( 1 + j x ) + l ( 1 − j x ) 0 0 t − b 2 n t − b t ( 1 + j y ) + b ( 1 − j y ) 0 0 0 − 1 − 2 n 0 0 − 1 0 ⎦ ⎤ Symmetric Vertical FOV ⎣ ⎡ w h cot ( 2 α ) 0 j x 0 0 cot ( 2 α ) j y 0 0 0 − 1 − 2 n 0 0 − 1 0 ⎦ ⎤ Symmetric Horizontal FOV ⎣ ⎡ cot ( 2 β ) 0 j x 0 0 h w cot ( 2 β ) j y 0 0 0 − 1 − 2 n 0 0 − 1 0 ⎦ ⎤
Jitter Perspective Transformation [ n , f ] \mathbf{\lbrack n,f \rbrack} [ n , f ] z ∈ [ 1 , − 1 ] \mathbf{z \in \lbrack 1,-1 \rbrack} z ∈ [ 1 , − 1 ]
Frustum Symmetric Vertical FOV Symmetric Horizontal FOV [ 2 n r − l 0 0 0 0 2 n t − b 0 0 r ( 1 + j x ) + l ( 1 − j x ) r − l t ( 1 + j y ) + b ( 1 − j y ) t − b f + n f − n − 1 0 0 2 f n f − n 0 ] [ h w cot ( α 2 ) 0 0 0 0 cot ( α 2 ) 0 0 j x j y f + n f − n − 1 0 0 2 f n f − n 0 ] [ cot ( β 2 ) 0 0 0 0 w h cot ( β 2 ) 0 0 j x j y f + n f − n − 1 0 0 2 f n f − n 0 ] \begin{array}{ccc} \textrm{Frustum} & \textrm{Symmetric Vertical FOV} & \textrm{Symmetric Horizontal FOV} \\ \begin{bmatrix} \frac{2n}{r-l} & 0 & 0 & 0\\ 0 & \frac{2n}{t-b} & 0 & 0\\ \frac{r(1+j_x)+l(1-j_x)}{r-l} & \frac{t(1+j_y)+b(1-j_y)}{t-b} & \frac{f+n}{f-n} & -1\\ 0 & 0 & \frac{2fn}{f-n} & 0\\ \end{bmatrix} & \begin{bmatrix} \frac{h}{w}\cot(\frac{\alpha}{2}) & 0 & 0 & 0\\ 0 & \cot(\frac{\alpha}{2}) & 0 & 0\\ j_x & j_y & \frac{f+n}{f-n} & -1\\ 0 & 0 & \frac{2fn}{f-n} & 0\\ \end{bmatrix} & \begin{bmatrix} \cot(\frac{\beta}{2}) & 0 & 0 & 0\\ 0 & \frac{w}{h}\cot(\frac{\beta}{2}) & 0 & 0\\ j_x & j_y & \frac{f+n}{f-n} & -1\\ 0 & 0 & \frac{2fn}{f-n} & 0\\ \end{bmatrix} \end{array} Frustum ⎣ ⎡ r − l 2 n 0 r − l r ( 1 + j x ) + l ( 1 − j x ) 0 0 t − b 2 n t − b t ( 1 + j y ) + b ( 1 − j y ) 0 0 0 f − n f + n f − n 2 f n 0 0 − 1 0 ⎦ ⎤ Symmetric Vertical FOV ⎣ ⎡ w h cot ( 2 α ) 0 j x 0 0 cot ( 2 α ) j y 0 0 0 f − n f + n f − n 2 f n 0 0 − 1 0 ⎦ ⎤ Symmetric Horizontal FOV ⎣ ⎡ cot ( 2 β ) 0 j x 0 0 h w cot ( 2 β ) j y 0 0 0 f − n f + n f − n 2 f n 0 0 − 1 0 ⎦ ⎤
Jitter Perspective Transformation infinite far plane [ n , ∞ ] \mathbf{\lbrack n,\infty \rbrack} [ n , ∞ ] z ∈ [ 1 , − 1 ] \mathbf{z \in \lbrack 1,-1 \rbrack} z ∈ [ 1 , − 1 ]
Frustum Symmetric Vertical FOV Symmetric Horizontal FOV [ 2 n r − l 0 0 0 0 2 n t − b 0 0 r ( 1 + j x ) + l ( 1 − j x ) r − l t ( 1 + j y ) + b ( 1 − j y ) t − b 1 2 n 0 0 − 1 0 ] [ h w cot ( α 2 ) 0 0 0 0 cot ( α 2 ) 0 0 j x j y 1 − 1 0 0 2 n 0 ] [ cot ( β 2 ) 0 0 0 0 w h cot ( β 2 ) 0 0 j x j y 1 − 1 0 0 2 n 0 ] \begin{array}{ccc} \textrm{Frustum} & \textrm{Symmetric Vertical FOV} & \textrm{Symmetric Horizontal FOV} \\ \begin{bmatrix} \frac{2n}{r-l} & 0 & 0 & 0\\ 0 & \frac{2n}{t-b} & 0 & 0\\ \frac{r(1+j_x)+l(1-j_x)}{r-l} & \frac{t(1+j_y)+b(1-j_y)}{t-b} & 1 & 2n\\ 0 & 0 & -1 & 0\\ \end{bmatrix} & \begin{bmatrix} \frac{h}{w}\cot(\frac{\alpha}{2}) & 0 & 0 & 0\\ 0 & \cot(\frac{\alpha}{2}) & 0 & 0\\ j_x & j_y & 1 & -1\\ 0 & 0 & 2n & 0\\ \end{bmatrix} & \begin{bmatrix} \cot(\frac{\beta}{2}) & 0 & 0 & 0\\ 0 & \frac{w}{h}\cot(\frac{\beta}{2}) & 0 & 0\\ j_x & j_y & 1 & -1\\ 0 & 0 & 2n & 0\\ \end{bmatrix} \end{array} Frustum ⎣ ⎡ r − l 2 n 0 r − l r ( 1 + j x ) + l ( 1 − j x ) 0 0 t − b 2 n t − b t ( 1 + j y ) + b ( 1 − j y ) 0 0 0 1 − 1 0 0 2 n 0 ⎦ ⎤ Symmetric Vertical FOV ⎣ ⎡ w h cot ( 2 α ) 0 j x 0 0 cot ( 2 α ) j y 0 0 0 1 2 n 0 0 − 1 0 ⎦ ⎤ Symmetric Horizontal FOV ⎣ ⎡ cot ( 2 β ) 0 j x 0 0 h w cot ( 2 β ) j y 0 0 0 1 2 n 0 0 − 1 0 ⎦ ⎤
Miscellaneous Matrices
Plane defined P : A x + B y + C z + D = 0 P: Ax + By + Cz + D = 0 P : A x + B y + C z + D = 0
Light position L : ( L x , L y , L z ) L: (L_x, L_y, L_z) L : ( L x , L y , L z )
where w = 0 w = 0 w = 0
w = 1 w = 1 w = 1
L ⃗ = ( L x , L y , L x , w ) \vec{L} = (L_x, L_y, L_x, w) L = ( L x , L y , L x , w ) P ⃗ = ( A , B , C , D ) \vec{P} = (A, B, C, D) P = ( A , B , C , D ) k = P ⃗ ⋅ L ⃗ k = \vec{P} \cdot \vec{L} k = P ⋅ L [ k − P ⃗ x L ⃗ x − P ⃗ y L ⃗ x − P ⃗ z L ⃗ x − P ⃗ d L ⃗ x − P ⃗ x L ⃗ y k − P ⃗ y L ⃗ y − P ⃗ z L ⃗ y − P ⃗ d L ⃗ y − P ⃗ x L ⃗ z − P ⃗ y L ⃗ z k − P ⃗ z L ⃗ z − P ⃗ d L ⃗ z − P ⃗ x L ⃗ w − P ⃗ y L ⃗ w − P ⃗ z L ⃗ w k − P ⃗ d L ⃗ w ] \begin{bmatrix} k - \vec{P}_x \vec{L}_x & -\vec{P}_y \vec{L}_x & -\vec{P}_z \vec{L}_x & -\vec{P}_d \vec{L}_x \\ -\vec{P}_x \vec{L}_y & k - \vec{P}_y \vec{L}_y & -\vec{P}_z \vec{L}_y & -\vec{P}_d \vec{L}_y \\ -\vec{P}_x \vec{L}_z & -\vec{P}_y \vec{L}_z & k - \vec{P}_z \vec{L}_z & -\vec{P}_d \vec{L}_z \\ -\vec{P}_x \vec{L}_w & -\vec{P}_y \vec{L}_w & -\vec{P}_z \vec{L}_w & k - \vec{P}_d \vec{L}_w \\ \end{bmatrix} ⎣ ⎡ k − P x L x − P x L y − P x L z − P x L w − P y L x k − P y L y − P y L z − P y L w − P z L x − P z L y k − P z L z − P z L w − P d L x − P d L y − P d L z k − P d L w ⎦ ⎤
