Perspective projection

Remarks

Example: the two rails $(1, -1, t)$ and $(-1, -1, t)$, $t > 0$, are parallel lines in the direction $(0, 0, 1)$. Their images $(f/t, -f/t)$ and $(-f/t, -f/t)$ lie on the lines $y = -x$ and $y = x$, which meet at $(0, 0)$, and both images tend to $(0, 0)$ as $t \to \infty$. So $(0, 0)$ is the vanishing point of the direction $(0, 0, 1)$; in general the vanishing point of a direction $v$ with $v_3 \ne 0$ is the image of $v$ itself, $(f v_1 / v_3, f v_2 / v_3)$. Non-example: the orthographic projection $(x, y, z) \mapsto (x, y)$ is not a perspective projection, since its projecting lines are parallel and do not pass through one centre. It maps parallel lines to parallel lines. The points of a line not through $O$ that lie in the domain map to points of one line, and lines parallel to $\pi$ keep their direction. In homogeneous coordinates the map is linear: $[x : y : z : 1] \mapsto [f x : f y : z]$, which is how graphics systems implement it.