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  "body": "\n## Definition\n\nLet $E$ be Euclidean 3-space, let $O \\in E$ be a point, the centre, and let $\\pi$ be a [[plane|0V3XAMF3C1]] not containing $O$, the picture plane.\nLet $P \\ne O$ be a point such that the [[line|0MP9BCF6DP]] $OP$ is not parallel to $\\pi$.\nThe *perspective projection* from $O$ onto $\\pi$ sends $P$ to the point where $OP$ meets $\\pi$.\nIn coordinates with $O = (0, 0, 0)$ and $\\pi = \\set{z = f}$ for some $f \u003e 0$, it is\n$$(x, y, z) \\mapsto \\left( \\frac{f x}{z}, \\frac{f y}{z} \\right), \\qquad z \\ne 0.$$\n\n## Remarks\n\nExample: the two rails $(1, -1, t)$ and $(-1, -1, t)$, $t \u003e 0$, are parallel lines in the direction $(0, 0, 1)$.\nTheir 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 $t \\to \\infty$.\nSo $(0, 0)$ 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, $(f v_1 / v_3, f v_2 / v_3)$.\nNon-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.\nIt maps parallel lines to parallel lines.\nThe 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.\nIn [[homogeneous coordinates|FS9FRJ17D2]] the map is linear: $[x : y : z : 1] \\mapsto [f x : f y : z]$, which is how graphics systems implement it.\n",
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