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  "body": "\n## Definition\n\nLet $F$ be a [[field|04C5ZBG1WT]] and let $\\mathbb{P}^2(F)$ be the [[projective plane|ZT9A6JK8QF]] over $F$, with homogeneous coordinates $[x : y : z]$.\nLet $Q$ be a nonzero [[quadratic form|0RBQS1JJCR]] on $F^3$.\nThe *projective conic* of $Q$ is the set\n$$C(Q) = \\set{[x : y : z] \\in \\mathbb{P}^2(F) : Q(x, y, z) = 0}.$$\nThe form $Q$ and its conic are *non-degenerate* if no invertible linear change of coordinates turns $Q$ into a form in at most two variables.\n\n## Remarks\n\nExample: $Q(x, y, z) = xz - y^2$.\nIts points are $[1 : t : t^2]$ for $t \\in F$ and the point $[0 : 0 : 1]$: if $x \\ne 0$ scale to $x = 1$ and get $z = y^2$; if $x = 0$ then $y = 0$.\nIt is non-degenerate.\nIf $Q$ involved at most two variables in some coordinates, there would be a nonzero vector $v$ with $Q(u + v) = Q(u)$ for all $u$.\nHere $Q(u + v) - Q(u) = u_x v_z + u_z v_x - 2 u_y v_y + Q(v)$, and this vanishes for all $u$ only if $v_x = v_z = 0$ and $Q(v) = -v_y^2 = 0$, so $v = 0$.\nIf $F = \\F_q$, this conic has $q + 1$ points.\n\nNon-example of non-degeneracy: $Q(x, y, z) = xy$ gives the union of the two lines $x = 0$ and $y = 0$.\nIt involves only $x$ and $y$, so it is degenerate.\n\nOver $\\R$, the non-degenerate form $x^2 + y^2 + z^2$ has an empty conic, so the form carries more information than the point set.\nOver an algebraically closed field, or over $\\R$ when the conic has a point, all non-degenerate conics are equivalent under projective transformations.\nThe affine [[conic sections|00HJCMC2CD]] are the parts of projective conics off the line at infinity.\n",
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