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  "body": "\n## Definition\n\nLet $\\Pi$ be a [[projective plane|DANFNN8C4Y]].\nA line is a *tangent* of a set $O$ of points at $P \\in O$ if it meets $O$ only in $P$.\nA set $O$ of points of $\\Pi$ is an *oval* if:\n\n1. every line of $\\Pi$ meets $O$ in at most two points;\n2. for every point $P \\in O$ there is exactly one tangent of $O$ at $P$.\n\n## Remarks\n\nIf $\\Pi$ is finite of [[order|K2YVMTGTBB]] $n$, a set $O$ is an oval if and only if $O$ has $n + 1$ points and no three of them on a line.\nIndeed, suppose $O$ has no three points on a line, and let $P \\in O$.\nOf the $n + 1$ lines through $P$, exactly $\\abs{O} - 1$ meet $O$ again.\nSo $P$ has $n + 2 - \\abs{O}$ tangents, and this equals $1$ exactly when $\\abs{O} = n + 1$.\n\nExample: in $PG(2, q)$ the non-degenerate [[conic|R3TMVHTP3A]] $xz = y^2$ has the $q + 1$ points $[1 : t : t^2]$, $t \\in \\F_q$, and $[0 : 0 : 1]$.\nA line $ax + by + cz = 0$, with $(a, b, c) \\ne 0$, meets it in the points $[1 : t : t^2]$ with $a + bt + ct^2 = 0$, and in $[0 : 0 : 1]$ exactly when $c = 0$.\nIf $c \\ne 0$, the polynomial has at most two roots.\nIf $c = 0$, it has at most one root, and the line also contains $[0 : 0 : 1]$.\nIn both cases the line meets the conic in at most two points.\nSo the conic is a set of $q + 1$ points with no three on a line, hence an oval.\n\nNon-example: three non-collinear points $A, B, C$ in $PG(2, 3)$.\nNo line contains all three, so clause 1 holds.\nThrough $A$ there are $4$ lines, and only $AB$ and $AC$ meet the set again.\nSo $A$ has two tangents, and clause 2 fails.\n",
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        "url": "https://en.wikipedia.org/wiki/Oval_(projective_plane)"
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