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  "body": "\n## Definition\n\nLet $\\ell$ and $m$ be two distinct lines of a [[projective plane|DANFNN8C4Y]], and let $O$ be a point that lies on neither of them.\nThe *perspectivity* from $\\ell$ to $m$ with centre $O$ sends each point $X$ of $\\ell$ to the common point of the lines $OX$ and $m$.\n\n## Remarks\n\nThe map is a bijection from the points of $\\ell$ to the points of $m$; its inverse is the perspectivity from $m$ to $\\ell$ with the same centre.\nIt fixes the common point of $\\ell$ and $m$.\nA composition of finitely many perspectivities is a *projectivity*.\nDually, let $A$ and $B$ be points and $a$ a line through neither.\nThe perspectivity with axis $a$ sends each line $x$ through $A$ to the line through $B$ and the point $x \\cap a$.\n\nExample: in the real affine plane, completed to $\\mathbb{P}^2(\\R)$, let $\\ell$ be $y = 1$, let $m$ be $y = 2$, and let $O = (0, 0)$.\nThe perspectivity sends $(t, 1)$ to $(2t, 2)$.\nCheck: the line through $O$ and $(t, 1)$ is $\\set{(st, s)}$, and it meets $y = 2$ at $s = 2$.\nThe common point of $\\ell$ and $m$ is their point at infinity, which is fixed.\n\nNon-example: if the centre $O$ lies on $\\ell$, the rule fails at $X = O$, since there is no line $OX$ for $X = O$.\n",
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