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I am trying to find an equation for a camera projection. The goal is to map the point $P(P_x, P_y, P_z)$ from the world coordinates onto the window coordinates $Q(Q_x, Q_y)$.

enter image description here

The eye of viewing is located a distance D from the $\hat{e_1}$-$\hat{e_2}$ plane, and in the direction of -$\hat{e_3}$. $\hat{e_1}$, $\hat{e_2}$ and $\hat{e_3}$ are unit vectors.

What is the equation to map P from world coordinates to window coordinates? $\vec{Q}(Q_x, Q_y) = f(\vec{P}, \hat{e_1}, \hat{e_2}, \hat{e_3}, \vec{v}, \vec{c})$ with all the vectors on the right side described in terms of world coordinates, eg

$\vec{P} = (P_x, P_y, P_z)$

$\vec{C} = (C_x, C_y, C_z)$

$\vec{e_1} = ({e_1}_x, {e_1}_y, {e_1}_z)$

$\vec{e_2} = ({e_2}_x, {e_2}_y, {e_2}_z)$

$\vec{e_3} = ({e_3}_x, {e_3}_y, {e_3}_z)$

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    $\begingroup$ If you don't know where to start, you can work on a simpler version of the problem: instead of projecting from a 3D space to a 2D screen, try projecting from 2D space to a 1D screen. Work one transformation at a time, and the equation in 2D becomes reasonably simple. Then you can redo it in 3D. $\endgroup$
    – Julien Guertault
    Commented Aug 27, 2018 at 4:18
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    $\begingroup$ By the way, the convention is usually to name the components of a vector $\vec{V} = (V_x, V_y, V_z)$ rather than $\vec{V} = (x_v, y_v, z_v)$. $\endgroup$
    – Julien Guertault
    Commented Aug 27, 2018 at 13:49
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    $\begingroup$ @raindrop - Check this link might come handy. computergraphics.stackexchange.com/questions/6365/… $\endgroup$
    – gallickgunner
    Commented Aug 28, 2018 at 18:32

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This is pretty straightforward matrix math and can be found on Wikipedia. It's quite elaborate, though, if you want to learn the math behind it and the other types of projection that exist. I will encourage you to checkout either Real-Time Rendering or Computer Graphics, Principles and Practice.

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  • $\begingroup$ I accept that the math is elaborate and I will really have to put in the time to study it. $\endgroup$
    – raindrop
    Commented Aug 27, 2018 at 13:19

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