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  "body": "\n## Definition\n\nLet $\\Pi$ be a [[projective plane|DANFNN8C4Y]].\nFor a point $U$, let $B(U)$ be the pencil of all lines through $U$.\nFor a line $g$, let $R(g)$ be the range of all points on $g$.\nFor a point $U$ and a line $g$ not through $U$, the maps $B(U) \\to R(g)$, $m \\mapsto m \\cap g$, and its inverse $R(g) \\to B(U)$, $P \\mapsto PU$, are [[perspectivities|QTZWZR1KWQ]].\nA *projectivity* $B(U) \\to B(V)$ is a composition of finitely many such perspectivities.\nLet $U$ and $V$ be distinct points and let $\\pi \\colon B(U) \\to B(V)$ be a projectivity with $\\pi(UV) \\ne UV$.\nThe *Steiner conic* of $\\pi$ is the set of points\n$$C = \\set{m \\cap \\pi(m) : m \\in B(U)}.$$\n\n## Remarks\n\nThe condition $\\pi(UV) \\ne UV$ makes $m$ and $\\pi(m)$ distinct lines for every $m$, so each intersection is a single point.\nThe points $U$ and $V$ lie on $C$: $UV \\cap \\pi(UV) = V$, and the line $m$ with $\\pi(m) = UV$ gives $m \\cap UV = U$.\n\nExample: in the [[projective plane|ZT9A6JK8QF]] over a field $F$, let $U = [1 : 0 : 0]$ and $V = [0 : 0 : 1]$.\nThe lines through $U$ are $by + cz = 0$ and the lines through $V$ are $ax + by = 0$, with $(b, c)$ and $(a, b)$ up to scaling.\nLet $\\pi$ send $by + cz = 0$ to $bx + cy = 0$; this map is linear in the parameters, and over a field such maps are projectivities.\nThe line $UV$ is $y = 0$, and $\\pi$ sends it to $x = 0$, so $\\pi(UV) \\ne UV$.\nSolving $by + cz = 0$ and $bx + cy = 0$ with $t = -c/b$ gives the points $[t^2 : t : 1]$, and $b = 0$ gives $U$.\nSo $C$ is the conic $xz = y^2$.\n\nNon-example: let $a$ be a line through neither $U$ nor $V$, and let $\\pi(m)$ be the line joining $V$ to $m \\cap a$.\nThen $\\pi(UV) = UV$, so the condition fails.\nFor $m \\ne UV$ the point $m \\cap \\pi(m)$ is $m \\cap a$, so these points are the points of the line $a$ other than $a \\cap UV$, not a conic.\n\nIn a plane over a field, Steiner's theorem says that the Steiner conics are exactly the non-degenerate [[projective conics|R3TMVHTP3A]].\n",
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