For questions related to Cayley–Klein metric. It is a metric on the complement of a fixed quadric in a projective space which is defined using a cross-ratio.
In mathematics, a Cayley–Klein metric is a metric on the complement of a fixed quadric in a projective space which is defined using a cross-ratio.
Suppose that $Q$ is a fixed quadric in projective space that becomes the absolute of that geometry. If $a$ and $b$ are $2$ points then the line through a and b intersects the quadric $Q$ in two further points $p$ and $q$. The Cayley–Klein distance $d(a,b)$ from $a$ to $b$ is proportional to the logarithm of the cross-ratio:
$$\displaystyle d(a,b)=C\log {\frac {\left|bp\right|\left|qa\right|}{\left|ap\right|\left|qb\right|}}$$
for some fixed constant C.
When C is real, it represents the hyperbolic distance of hyperbolic geometry, when imaginary it relates to elliptic geometry.
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