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  "body": "\n## Definition\n\nLet $\\Pi$ be a finite [[projective plane|DANFNN8C4Y]] and let $k \\ge 1$ be an integer.\nA *$k$-arc* of $\\Pi$ is a set of $k$ points of $\\Pi$ such that no three of them lie on a common line.\n\n## Remarks\n\nLet $\\Pi$ have [[order|K2YVMTGTBB]] $n$, so each point lies on $n + 1$ lines.\nIf $P$ is a point of a $k$-arc $K$, the lines joining $P$ to the other $k - 1$ points of $K$ are distinct.\nHence $k - 1 \\le n + 1$, that is, $k \\le n + 2$.\nWhen $n$ is odd, $k \\le n + 1$.\nAn arc with $n + 1$ points is called an oval, and one with $n + 2$ points a hyperoval.\n\nExample: in the [[plane $\\mathrm{PG}(2, q)$|CSMB49BDPT]], let $K$ consist of the points $(1 : t : t^2)$ for $t \\in \\F_q$ and the point $(0 : 0 : 1)$.\nThree points $(1 : s : s^2)$, $(1 : t : t^2)$, $(1 : u : u^2)$ with distinct $s, t, u$ are not collinear, because their Vandermonde determinant $(t - s)(u - s)(u - t)$ is nonzero.\nFor $(1 : s : s^2)$, $(1 : t : t^2)$ and $(0 : 0 : 1)$ the determinant is $t - s \\ne 0$.\nSo $K$ is a $(q + 1)$-arc.\n\nNon-example: the $n + 1$ points of a line, for $n \\ge 2$, contain three collinear points, so they do not form an arc.\n\nMore generally, a $(k, d)$-arc is a set of $k$ points with at most $d$ on any line, so that a $k$-arc is a $(k, 2)$-arc.\n",
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