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  "body": "\n## Definition\n\nLet $E$ be Euclidean 3-space with [[Cartesian coordinates|0RWKYX8339]] $(x, y, z)$.\nLet $\\pi_1$ be the horizontal [[plane|0V3XAMF3C1]] $z = 0$ and $\\pi_2$ the vertical plane $y = 0$; they meet in the ground line $x_{12}$, the $x$-axis.\nThe *Monge projection* of a point $P = (x, y, z)$ is the pair of its orthogonal projections, the top view $P' = (x, y, 0) \\in \\pi_1$ and the front view $P'' = (x, 0, z) \\in \\pi_2$.\nRotate $\\pi_2$ about $x_{12}$ onto $\\pi_1$, so that both views lie in one drawing plane.\nThen $P'$ and $P''$ lie on a common line perpendicular to $x_{12}$.\n*Descriptive geometry* is the study of figures of $E$ through such pairs of views, or through more views on further image planes.\nIts aim is to solve spatial problems by plane constructions on the views.\n\n## Remarks\n\nThe map $P \\mapsto (P', P'')$ is injective, since the two views give $x$, $y$ and $z$.\nSo a pair of views on a common perpendicular to $x_{12}$ determines a unique point.\nExample: for a segment $PQ$ the true length follows from the views, since $\\abs{PQ}^2 = \\abs{P'Q'}^2 + (z_P - z_Q)^2$, where $\\abs{P'Q'}$ is measured in the top view and $z_P - z_Q$ in the front view.\nFor $P = (0, 0, 0)$ and $Q = (3, 4, 12)$, one gets $\\abs{P'Q'} = 5$ and $\\abs{PQ} = \\sqrt{25 + 144} = 13$.\nNon-example: a single view does not determine the point: the points $(1, 2, 0)$ and $(1, 2, 5)$ have the same top view.\nThe [[Euclidean space|0XPR1E85W0]] can also be shown by central projections, as in a [[perspective projection|YHKSDF59HK]].\n",
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