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  "body": "\n## Summary\n\nA Cayley–Klein metric is a distance on a region of [[projective space|ZT9A6JK8QF]].\nIt is built from a fixed [[quadric|Q3Q4FF23VG]], the absolute, through the logarithm of a [[cross-ratio|09ASMRMD67]].\nDifferent choices of the absolute give [[hyperbolic|GN4N94GQ0Z]] and [[elliptic|H5GB19W2H3]] geometry, and Euclidean geometry appears as a limit case.\nThe construction of Cayley and Klein showed that these metric geometries all live inside [[projective geometry|N32Z68F0D3]].\n\n## Explanations\n\n### L1 human\n\nFix an ellipse in the plane and look only at the points inside it.\nTo measure the distance between two inside points, draw the line through them and see where it hits the ellipse.\nThe distance is half the logarithm of a ratio of four lengths along that line, the cross-ratio.\nPoints near the ellipse are very far from everything, so the ellipse acts as a horizon at infinity.\nWith a circle as the absolute, this gives the [[Beltrami–Klein model|1WN9B18H86]] of the hyperbolic plane.\n",
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