Is it possible to determine the left/right plane equations of a frustum, given the near plane? If I am given four vertex points of the near plane that results from cutting the head of a pyramid off, is it possible to find the equation of the planes of the frustum to the left and right which are supposedly perpendicular to the near plane? Is it also possible to get the 4 vertex points of the far plane of the frustum if i know the field of view and height of the pyramid? I ask this because I am having trouble trying to get all 6 planes of the frustum, and all I have is just the information listed above. So I am trying to figure out a way to determine them starting from a simple approach.

* You have a point $P =\begin{pmatrix} x \\\ y \\\ z \end{pmatrix}$ in your 3d scene. Add a $1$ to get $P' = \begin{pmatrix} x \\\ y \\\ z \\\ 1 \end{pmatrix}$.

* Multiply $P'$ with the projection matrix $\scriptstyle\begin{pmatrix} 1 & 0 & 0 & 0 \\\ 0 & 1 & 0 & 0 \\\ 0 & 0 & 1 & 0 \\\ 0 & 0 & 1 & -d \end{pmatrix} $ to get $P'' = \begin{pmatrix} x\\\ y \\\ z \\\ z-d \end{pmatrix}$

* Divide $P''$ by $w''= z-d $ to get $P''' = \begin{pmatrix} x/(z-d) \\\ y/(z-d) \\\ z/(z-d) \\\ 1 \end{pmatrix}$

* If $x'',y'',z'' \in [-1,1]$, print the point at the position $(x'',y'')$ on the screen.




Can you find back the position of the 9 points of the frustrum from those transformation ?