Fork me on GitHub
Math for the people, by the people.

User login

M\"obius transformation cross-ratio preservation theorem

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

Mathematics Subject Classification

30E20 no label found

Comments

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:
=====
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
$$g(f(z_1),f(z_2),f(z_3),f(z_4))=gf^{-1}(f(z_1))=g(z_1)\;.$$
=====
Larry

Subscribe to Comments for "M\"obius transformation cross-ratio preservation theorem"