normal form of 4 points

Autor:Walter Füchte

this activity is a page of geogebra-book elliptic functions & bicircular quartics & . . .(06.02.2023)

move z₁, z₂, z₃, z₄

To 4 different points

there is a Möbius transformation ie.: a rational function of the form

which maps the points to 4 complex image points

, with a suitable

. The image points lie point-symmetrically to the pairs of points

. We call this position the normal form of the 4 points. If the points

are concyclic, the image points lie on one of the axes or on the unit circle. If the points are mirror-inverted on 2 orthogonal circles, the image points are also mirror-inverted on the angle bisectors. You can then also place them mirror-inverted on the axes - this position is also called normal form. If the original points have a harmonic position, the image points are the intersections of the unit circle with the angle bisectors. For the proof we construct 3 pairwise orthogonal circles in such a way that the original points lie point-symmetrically to the intersection point pairsin the sense of Möbius geometry.Geometrically, this construction is quite complex. Mathematically, one finds the connection with the help of the LIE algebra of the Möbius group by a very simple and short calculation (see below!). The actual idea for both the construction and the calculation is based on the principle of repeated symmetrisation! For the construction: The 4 points can be divided into 2 pairs of points in 3 different ways. For example, determine the circles

and their bisectors

and

, as well as the circles

with the corresponding angle bisector circles

and

for the pairs of points

. 2 of these 4 angle bisector circles do not intersect; to the other two circles you construct their angle bisector circles (yellow in the applet!). This construction can be done in three different ways, whereby for 2 different ways, 2 of the resulting angle bisector circles are identical! The result is 3 orthogonal circles. Their intersection point pairs are mapped with a Möbius transformation to the given pairs of points

and

. For the calculation: 2 points (e.g.

) determine an elliptical pencil of circles whose circles can be seen as trajectories of a Möbius motion. The corresponding infinitesimal movement is briefly indicated as

. Accordingly, let

be interpreted. The LIE-product

belongs to an infinitesimal movement,i.e. elliptical circular motion, whose pencil points lie harmonically with those of

and

. The LIE-products

are pairwise "orthogonal", which geometrically means that the base point pairs lie on 3 pairwise orthogonal circlesand the 4 given points lie harmonically, i.e. point-symmetrically, to these base point pairs!

normal form of 4 points

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