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  "body": "\n## Definition\n\nLet $\\Pi$ be a [[projective plane|DANFNN8C4Y]] and let $\\ell$ be a line of $\\Pi$.\nAn *elation with axis $\\ell$* is a [[collineation|V7PRPN24CV]] $\\alpha$ of $\\Pi$ with two properties.\nIt fixes every point of $\\ell$, and for some point $C$ on $\\ell$ it maps every line through $C$ to itself.\nThe plane $\\Pi$ is a *translation plane with respect to $\\ell$* if the following holds.\nFor any two points $P$ and $Q$ not on $\\ell$, some elation with axis $\\ell$ maps $P$ to $Q$.\nThe line $\\ell$ is then a *translation line* of $\\Pi$.\n\n## Remarks\n\nExample: let $F$ be a field and let $\\Pi$ be the projective plane over $F$, with $\\ell$ the line at infinity.\nFor $(a, b) \\in F^2$, the translation $(x, y) \\mapsto (x + a, y + b)$ of $F^2$ maps lines to parallel lines, so it extends to a collineation of $\\Pi$ that fixes every point at infinity.\nIf $(a, b) \\ne (0, 0)$, it maps each line of direction $(a, b)$ to itself.\nSo it fixes every line through the point at infinity $C$ of that direction, and it is an elation with axis $\\ell$.\nThe translation by $Q - P$ maps $P$ to $Q$, so $\\Pi$ is a translation plane with respect to $\\ell$.\n\nNon-example: the Hughes plane of order $9$ is a projective plane that is not a translation plane with respect to any of its lines.\n\nWhen $\\Pi$ is a translation plane with respect to $\\ell$, the elations with axis $\\ell$ form an abelian group that acts regularly on the points off $\\ell$.\nThe translation planes are exactly the planes coordinatized by [[quasifields|JQ22QH86DE]].\n",
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    "name": "Translation plane",
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