Perspective Projection

Perspective projection projects objects on a plane. It has the effect that distant objects are smaller.

projection

If we look at this scene in the negative x axis direction.

projection

Perspective matrix is being used to transform all verteices to the near clipping plane. For example, the picture above has a vertex(x, y, z). After the perspective matrix is applied, we will get projected vertex(x’, y’, z’). The coordinates of projected vertex can be calculated based on similar triangles.

$\frac{y'}{y} = \frac{near}{-z}$

z is inversely propotional to y since near is always positive and z is negative(camera is looking at negative z direction).

$y' = \frac{near\cdot y}{-z}$

Same idea can also be applied to x’

$x' = \frac{near\cdot x}{-z}$

However, it’s impossible to construct a 3 by 3 matrix that can transforms x component in vector (x, y, z) into near multiply x divided by z.

Homogeneous Coordinates

In homogeneous coordinates, a new component w is appended. And any non zero scaler multiply with the point represent the same point.

$\begin{bmatrix} x\\ y\\ z\\ w\\\end{bmatrix}= \begin{bmatrix} \frac{1}{w}x\\ \frac{1}{w}y\\ \frac{1}{w}z\\ 1\\\end{bmatrix} = \begin{bmatrix} 2x\\ 2y\\ 2z\\ 2w\\\end{bmatrix}$

If we assume the viewing plane is at w = 1. Then all points represented by homogeneous coordinates can be divided by w in order to get the projected point on the viewing screen. In other word, if perspective matrix can output a vector that w = z. Then projected vector position can be retrieved by dividing w after perspective matrix is applied.

After applying perspective matrix:

$\begin{bmatrix} nx\\ ny\\ z\\ -z\\\end{bmatrix}$

After dividing by w component:

$\begin{bmatrix} \frac{nx}{-z}\\ \frac{ny}{-z}\\ -1\\ 1\\\end{bmatrix}$

Therefore, the perspective matrix:

$\begin{bmatrix} n& 0& 0& 0\\ 0& n& 0& 0\\ 0& 0& 1& 0\\ 0& 0& -1& 0\\\end{bmatrix}$

Homogeneous Division:

$\begin{bmatrix} \frac{nx}{z}\\ \frac{ny}{z}\\ 1\\ 1\\\end{bmatrix}=\begin{bmatrix} nx\\ ny\\ z\\ z\\\end{bmatrix}=\begin{bmatrix} n& 0& 0& 0\\ 0& n& 0& 0\\ 0& 0& 1& 0\\ 0& 0& -1& 0\\\end{bmatrix}\begin{bmatrix} x\\ y\\ z\\ 1\\\end{bmatrix}$

Perspective Matrix

The problem of this matrix is the lost of depth information during homogeneous division(since w = -z).
If I want to keep depth information, it will be great if z component is -z^2 before homogeneous division.

$\begin{bmatrix} nx\\ ny\\ z^2\\ z\\\end{bmatrix}=\begin{bmatrix} n& 0& 0& 0\\ 0& n& 0& 0\\ 0& 0& m_{1}& m_{2}\\ 0& 0& 1& 0\\\end{bmatrix}\begin{bmatrix} x\\ y\\ z\\ 1\\\end{bmatrix}$

That gives us:

$zm_{1}+m_{2}=z^2$

We need 2 different z values to solve this equation.

frustum

The viewing region(also called frustum) has 2 depth restrictions: near and far. we use n and f to represent these 2 values.
n and f are 2 z values which can be used to solve the equation above.
The solutions of m1 and m2 are:

$m_{1} = n+f$

$m_{2} = nf$

The perspective matrix:

$\begin{bmatrix} n& 0& 0& 0\\ 0& n& 0& 0\\ 0& 0& n+f& -nf\\ 0& 0& -1& 0\\\end{bmatrix}$

Perpective Projection Matrix

After applying perspective transformation and homogeneous divison, the frsutum becomes a axis ligned bounding box.

frustum_to_AABB

After that, we can apply orthographic projection matrix which transforms the viewing scene(inside AABB) into canonical view volume.

canonical_view

It’s worth mentioning that homogeneous divison is a mutiplication between a scalar(w) and a matrix.

$\frac{1}{w} \begin{bmatrix}M_{p}\end{bmatrix}\begin{bmatrix}x \\ y\\ z\\ w\\\end{bmatrix}$

Homogeneous division can be extracted(scalar factorization), perspective matrix and orthographic projection matrix can be applied first.

$\frac{1}{w}(\begin{bmatrix}M_{ortho}\end{bmatrix} \begin{bmatrix}M_{pers}\end{bmatrix})\begin{bmatrix}x \\ y\\ z\\w\\\end{bmatrix} = \begin{bmatrix}M_{ortho}\end{bmatrix} (\frac{1}{w}\begin{bmatrix}M_{pers}\end{bmatrix} ) \begin{bmatrix} x\\ y\\ z\\w\\\end{bmatrix}$

$\frac{1}{w}(\begin{bmatrix}M_{ortho}\end{bmatrix} \begin{bmatrix}M_{pers}\end{bmatrix})\begin{bmatrix} x\\ y\\ z\\ w\\\end{bmatrix} = \frac{1}{w}(\begin{bmatrix} \frac{2}{r-l}& 0& 0& -\frac{r+l}{r-l}\\ 0& \frac{2}{t-b}& 0& -\frac{t+b}{t-b}\\ 0& 0& -\frac{2}{n-f}& \frac{n+f}{n-f}\\ 0& 0& 0& 1\\\end{bmatrix}\begin{bmatrix} n& 0& 0& 0\\ 0& n& 0& 0\\ 0& 0& -(n+f)& nf\\ 0& 0& -1& 0\\\end{bmatrix})\begin{bmatrix} x\\ y\\ z\\ w\\\end{bmatrix}$

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

Perpective Projection Matrix:

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

Credits:
Learn WebGL
OpenGL Projection Matrix by songho
video by Brendan Galea, really good explanation

Perspective Projection

Derivation of Perspective Projection Matrix

Homogeneous Coordinates

Perspective Matrix

Perpective Projection Matrix

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