The location of 4 points

Author:Walter Füchte

this activity is a page of geogebra-book elliptic functions & bicircular quartics & . . .(27.04.2023)

this activity is also a page of GeoGebrabook Moebiusebene (29.09.2020)

Four (different) points *)

can be connected in three ways in pairs by straight lines: ( *) see the

activity before )

For the LIE-products

and

can easily be calculated as follows (with the help of the general development rule, calculation example see below **)):

If one thinks of the connecting line vectors as normalised from the outset:

the following apply

and for the LIE-products

and

(defined accordingly) is thus obtained:

. These straight line vectors therefore form in

an ON-Basis. The poles of these basis vectors are the intersections of 3 pairwise orthogonal circles and can, with suitable orientation, be mapped by a Möbius transformation to the point pairs

and

. Becaue of

follows

, thus the poles

and

from

separate the point pairs

and

harmonically. Be

the poles of

after the Möbius transformation, then - the images

resp.

are point-symmetrical to the origin:

and

. - The images of the point pairs

resp..

are correspondingly harmonic to

, i.e.

and

must apply. Conclusion: 4 different points of the Möbius plane can always be mapped by a Möbius transformation to

, for a suitable one

. We call this the representation of the 4 points in normal form. . Note on the underlying calculations for the applet above: the complex numbers

are mapped in the Euclidean KOS to the complex touch line vectors

. LIE products are calculated with the complex cross product. To calculate the poles of a straight line vector

The location of 4 points

New Resources

Discover Resources

Discover Topics