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  "body": "\n## Summary\n\nLie sphere geometry studies oriented spheres, oriented hyperplanes and points of $\\R^n$ on an equal footing.\nIts transformations are those that preserve oriented contact.\nThese objects correspond to the points of a [[quadric|Q3Q4FF23VG]] in projective space $\\mathbb{RP}^{n+2}$, the Lie quadric, given by a quadratic form of signature $(n+1, 2)$.\nOriented contact becomes orthogonality for that form, and the Lie sphere transformations are the projective maps preserving the quadric.\nIt contains the Möbius geometry of [[inversive geometry|HFBW3SXQVM]] and Laguerre geometry as subgeometries, and it is named after Sophus Lie.\n\n## Explanations\n\n### L1 human\n\nIn ordinary geometry, points, lines and circles are different kinds of objects.\nLie sphere geometry treats them all as circles: a point is a circle of radius zero and a line is a circle of infinite radius.\nGive each circle an orientation, a direction of travel, and call two oriented circles in contact when they touch with matching directions at the touching point.\nThe geometry keeps track only of which circles touch which; its transformations may turn points into circles and circles into lines, as long as touching is preserved.\n",
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