Perspective Projection
Extend the idea of homogeneous coordinates to 3D.
Perspective Projection
The perspective projection is a projection from 3D to 2D, so we need a 4 x 4 matrix to transform 3D points in homogeneous coordinates.
consider a horizontal cross section of the scene:
![x-section]()
x-section
The relationship between the the 3D camera coordinate \(x_c\) and the 2D image coordinate \(x_i\) is:
![x-section]()
x-section
\[\frac{x_i}{d} = \frac{x_c}{z_c}\]
\[\Rightarrow x_i = \frac{x_c}{\frac{z_c}{d}}\]
The relationship between the the 3D camera coordinate \(y_c\) and the 2D image coordinate \(y_i\) is:
![y-section]()
y-section
\[\frac{y_i}{d} = \frac{y_c}{z_c}\]
\[\Rightarrow y_i = \frac{y_c}{\frac{z_c}{d}}\]
Perspective Projection
We first extend the 3x3 homogeneous matrix for 2D graphics to a 4x4 matrix for 3D graphics.
A perspective projection from 3D to 2D can then be expressed as:
\[
\begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix} =
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 1 / d & 0
\end{bmatrix}
\begin{bmatrix}
x_c \\ y_c \\ z_c \\ 1
\end{bmatrix}
\]
Let’s work out the matrix multiplication for each coordinate after projection:
\[
x = x_c ~, ~ y = y_c ~, ~ z = z_c ~, ~ w = \frac{z_c}{d}
\]
- However, these are homogenous coordinates, i.e. they are scaled by a factor of w.
- take note that \(w = \frac{z_c}{d}\)
To find the corresponding image coordinates, we divide:
\[
x_i = \frac{x}{w} = \frac{x_c}{\frac{z_c}{d}} ~,
y_i = \frac{y}{w} = \frac{y_c}{\frac{z_c}{d}} ~,
z_i = \frac{z}{w} = \frac{z_c}{\frac{z_c}{d}} = d,
\]
so all \(z_i\) are equal to \(d\).
Perspective Projection
\[
\begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix} =
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 1 / d & 0
\end{bmatrix}
\begin{bmatrix}
x_c \\ y_c \\ z_c \\ 1
\end{bmatrix}
\]
