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  "body": "\n## Definition\n\nLet $(P, L)$ be a [[partial linear space|CX07CWPCDX]] with point set $P$ and line set $L$.\nTwo points are *collinear* if they are equal or lie on a common line.\nThe space $(P, L)$ is a *polar space* (in the sense of Buekenhout and Shult) if:\n\n1. every line has at least three points;\n2. for every line $\\ell$ and point $p$ not on $\\ell$, the point $p$ is collinear with exactly one or with all points of $\\ell$.\n\nThe polar space is *non-degenerate* if, in addition, no point is collinear with all points of $P$.\n\n## Remarks\n\nExample: let $F$ be a field and $Q(x) = x_0 x_1 + x_2 x_3$ on $F^4$, with polar form $B(x, y) = Q(x + y) - Q(x) - Q(y)$.\nLet $P$ be the points of the [[quadric|Q3Q4FF23VG]] of $Q$ in $\\mathbb{P}^3(F)$ and $L$ the projective lines contained in it.\nFor $p, q \\in P$ one has $Q(\\lambda p + \\mu q) = \\lambda \\mu B(p, q)$, so $p$ and $q$ are collinear exactly when $B(p, q) = 0$.\nFor a line $\\ell \\in L$ and $p \\in P$ off $\\ell$, the points $q$ of $\\ell$ with $B(p, q) = 0$ form a linear condition on $\\ell$, so they are one point or all of $\\ell$.\nEvery line has $\\abs{F} + 1 \\ge 3$ points.\nThe points collinear with $p$ lie in the plane $B(p, \\cdot) = 0$, which does not contain all of $P$, so the polar space is non-degenerate.\n\nNon-example: a projective plane with its lines satisfies clauses 1 and 2, since any two points are collinear.\nIt is degenerate, because every point is collinear with all points.\n\nThe rank of a non-degenerate polar space is the largest $n$ such that there is a chain of $n$ nonempty subspaces in which every two points are collinear.\nThe non-degenerate polar spaces of rank $2$ are exactly the generalized quadrangles with at least three points on each line.\nVeldkamp and Tits showed that the non-degenerate polar spaces of rank at least $3$ come from forms on vector spaces, with some exceptions in rank $3$.\nThe forms needed here include pseudo-quadratic forms.\n",
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