## You are here

HomeM\"obius transformation cross-ratio preservation theorem

## Primary tabs

# Möbius transformation cross-ratio preservation theorem

A Möbius transformation $f:z\mapsto w$ preserves the cross-ratios, i.e.

$\frac{(w_{1}-w_{2})(w_{3}-w_{4})}{(w_{1}-w_{4})(w_{3}-w_{2})}=\frac{(z_{1}-z_{% 2})(z_{3}-z_{4})}{(z_{1}-z_{4})(z_{3}-z_{2})}$ |

Conversely, given two quadruplets which have the same cross-ratio, there exists a Möbius transformation which maps one quadruplet to the other.

A consequence of this result is that the cross-ratio of $(a,b,c,d)$ is the value at $a$ of the Möbius transformation that takes $b$, $c$, $d$, to $1$, $0$, $\infty$ respectively.

Related:

CrossRatio

Major Section:

Reference

Type of Math Object:

Theorem

Parent:

## Mathematics Subject Classification

30E20*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections

## Comments

## 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:

=====

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