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  "body": "\n## Summary\n\nLaguerre geometry is the geometry of oriented lines and oriented circles of the plane.\nIts transformations preserve oriented contact and map oriented lines to oriented lines.\nPoints count as circles of radius $0$, and these transformations need not map points to points.\nA typical example moves every oriented line parallel to itself by the same distance and adds that distance to every signed radius.\nIt is a subgeometry of [[Lie sphere geometry|TFQSF9RA1G]], and its abstract incidence version is the [[Laguerre plane|F0JQ1TGKB0]]; it is named after Edmond Laguerre.\n\n## Explanations\n\n### L1 human\n\nThink of lines with arrows on them and circles with a direction of travel.\nA directed line and a directed circle touch properly when they are tangent and run the same way at the touching point.\nLaguerre geometry keeps only this touching relation and the fact of being a line.\nSo it may grow every circle by one centimetre and push every line one centimetre sideways at once.\nAll touching pairs still touch, even though a point has become a small circle.\n",
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