# M\"obius transformation cross-ratio preservation theorem

## Primary tabs

Major Section:
Reference
Type of Math Object:
Theorem
Parent:

## Mathematics Subject Classification

### cross-ratio

Hi,
If it were me I would rename this item "cross-ratio", and include
1) the proposition that the cross-ratio (a,b,c,d) is the
value at a of the mobius transformation that takes b,c,d, to
1,0,infty respectively
2) a proof of the preservation, something like:
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Write
$$g(z)=\frac{(z-z_2)(z_3-z_4)}{(z-z_4)(z_3-z_2)}\;.$$
The function $gf^{-1}$ takes $f(z_2),f(z_3),f(z_4)$ to 1,0,$\infty$
respectively. So, by the above characterization of the cross ratio,
we have
$$g(f(z_1),f(z_2),f(z_3),f(z_4))=gf^{-1}(f(z_1))=g(z_1)\;.$$
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Larry