# Cayley–Klein metric

The question recently arose in conversation whether a dissertation of 2 lines could deserve and get a Fellowship. ... Cayley's projective definition of length is a clear case if we may interpret "2 lines" with reasonable latitude. ... With Cayley the importance of the idea is obvious at first sight.

Littlewood (1986, pp. 39–40)

In mathematics, a Cayley–Klein metric is a metric on the complement of a fixed quadric in projective space defined using a cross-ratio. The first example was given by Cayley (1859), and they were studied further by Felix Klein (1871, 1873).

The Cayley–Klein metric can be used to define the distance in the Cayley–Klein model of hyperbolic geometry.

The Hilbert metric of a convex set is defined in a similar way.

## Definition

Suppose that Q is a fixed quadric in projective space. If p and q are 2 points then the line through p and q intersects the quadric Q in 2 further points a and b. The Cayley–Klein distance d(p,q) from p to q is proportional to the logarithm of the cross-ratio:

$d(p,q)=C \log \frac{|qa||bp|}{|pa||bq|}$

for some fixed constant C.