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  "body": "\n## Definition\n\nLet $F$ be a [[field|04C5ZBG1WT]], let $V$ be a [[vector space|000AYPA9AJ]] over $F$ of finite dimension $n + 1 \\ge 2$, and let $\\mathbb{P}(V)$ be its [[projective space|ZT9A6JK8QF]].\nLet $Q$ be a nonzero [[quadratic form|0RBQS1JJCR]] on $V$.\nFor a nonzero vector $v$, write $\\langle v \\rangle$ for the point of $\\mathbb{P}(V)$ that it spans.\nThe *(projective) quadric* of $Q$ is the set\n$$\\set{\\langle v \\rangle \\in \\mathbb{P}(V) : Q(v) = 0}.$$\nIt is well defined because $Q(\\lambda v) = \\lambda^2 Q(v)$.\nAn *affine quadric* in $F^n$ is the set of zeros of a polynomial of degree exactly $2$ in $n$ variables with coefficients in $F$.\n\n## Remarks\n\nExample: in $\\mathbb{P}^3(F)$, the form $Q(x) = x_0 x_1 + x_2 x_3$ defines a quadric that contains the line of points $[s : 0 : t : 0]$, since $Q(s, 0, t, 0) = 0$ for all $s, t$.\nExample: the sphere $x^2 + y^2 + z^2 = 1$ in $\\R^3$ is an affine quadric, and $x^2 + y^2 + z^2 - w^2 = 0$ is its projective closure.\n\nNon-example: the zero set of $y - x^3$ in $F^2$ is not an affine quadric, because the polynomial has degree $3$.\n\nFor $n = 2$ the projective quadrics are the [[projective conics|R3TMVHTP3A]].\nDifferent forms can define the same point set.\nFor example, over $\\R$ every positive definite form defines the empty quadric.\nFor this reason the quadric is often taken together with its form, up to a nonzero scalar.\n",
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