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  "body": "\n## Summary\n\nA translation plane is a [[projective plane|DANFNN8C4Y]] with a special line $\\ell$.\nThe collineations that act like translations with respect to $\\ell$ can move any point off $\\ell$ to any other.\nRemoving $\\ell$ leaves an affine plane with a full group of translations, as in the real plane.\nTranslation planes are exactly the planes coordinatized by [[quasifields|JQ22QH86DE]], and they form the largest well-understood family of non-Desarguesian planes.\n\n## Explanations\n\n### L1 human\n\nIn the ordinary plane one can slide every figure by a fixed vector, and these slides carry any point to any other point.\nA translation plane is a projective plane that keeps this property for one chosen line, which plays the role of the line at infinity.\nEach slide fixes every point at infinity and moves every line to a parallel line.\nThe real projective plane is a translation plane, and so is every finite plane $PG(2, q)$.\nThere are also translation planes that are not of this form, such as the Hall plane of order $9$.\n",
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