---
tag: J7ZE3EGRYX
kind: concept
name: Cayley–Klein metric
msc: [51F99, 51M10]
sources:
  - url: https://en.wikipedia.org/wiki/Cayley%E2%80%93Klein_metric
same_as:
  wikidata: Q5055329
  wikipedia: Cayley–Klein metric
---

## Summary

A Cayley–Klein metric is a distance on a region of [[projective space|ZT9A6JK8QF]].
It is built from a fixed [[quadric|Q3Q4FF23VG]], the absolute, through the logarithm of a [[cross-ratio|09ASMRMD67]].
Different choices of the absolute give [[hyperbolic|GN4N94GQ0Z]] and [[elliptic|H5GB19W2H3]] geometry, and Euclidean geometry appears as a limit case.
The construction of Cayley and Klein showed that these metric geometries all live inside [[projective geometry|N32Z68F0D3]].

## Explanations

### L1 human

Fix an ellipse in the plane and look only at the points inside it.
To measure the distance between two inside points, draw the line through them and see where it hits the ellipse.
The distance is half the logarithm of a ratio of four lengths along that line, the cross-ratio.
Points near the ellipse are very far from everything, so the ellipse acts as a horizon at infinity.
With a circle as the absolute, this gives the [[Beltrami–Klein model|1WN9B18H86]] of the hyperbolic plane.