Weak Perspective

This perspective model provides a form of 3D perspective, but the scaling happens between objects rather than across every point. Weak perspective hinges on an important assumption: the change in z (or depth) within a single object is not significant relative to its distance from the camera. For example, a person might be about a foot thick, but they are standing a mile from the camera.

All of the points in an object that is $z_0$ distance away from the projection plane will be mapped as such : $$(x, y, z) \rightarrow (\frac{fx}{z_0}, \frac{fy}{z_0})$$ This $z_0$ changes from object to object, so each object has its own scale factor. Its projection matrix is: $$\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1/s \end{bmatrix} \begin{bmatrix} x \\ y \\ z \\ 1/s \end{bmatrix} = \begin{bmatrix} x \\ y \\ 1/s \\ \end{bmatrix} \rightarrow (sx, sy)$$