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  "body": "\n## Definition\n\nLet $\\Pi$ be a finite [[projective plane|DANFNN8C4Y]] of [[order|K2YVMTGTBB]] $n$.\nA *hyperoval* of $\\Pi$ is a set of $n + 2$ points of $\\Pi$ such that no three of them lie on a line.\n\n## Remarks\n\nEvery line meets a hyperoval $H$ in $0$ or $2$ points.\nSuppose a line met $H$ in exactly one point $P$.\nThe other $n$ lines through $P$ would then cover the remaining $n + 1$ points of $H$, each at most one of them, which is impossible.\n\nExample: in the [[Fano plane|QQ8GW1ASZM]], of order $2$, let $H$ be the four points off a fixed line $\\ell$.\nAny other line meets $\\ell$ in one point, so it has exactly $2$ points in $H$; hence no three points of $H$ are collinear, and $\\abs{H} = 4 = 2 + 2$.\n\nNon-example: $PG(2, 3)$ has no hyperoval.\nMore generally, a plane of odd order $n$ has none.\nSuppose $S$ were a hyperoval, and take a point $P$ outside $S$.\nEvery line through $P$ meets $S$ in $0$ or $2$ points, so these lines split $S$ into pairs.\nThen $\\abs{S}$ is even, while $n + 2$ is odd.\n\nFor $n$ even, the tangents of an [[oval|7B03TB64N9]] all pass through one point, its nucleus, and adding the nucleus gives a hyperoval.\n",
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