the möbiusgroup SO(3, ℂ)

Author:Walter Füchte

Topic:Circle, Complex Numbers, Differential Equation, Exponential Functions, Geometric Transformations

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)

In the complex, 3-dimensional vector space

with non-degenerate quadratic form

an oriented basis is selected

with

, for which the two product tables are to apply:

The name

is chosen because this vector space can be interpreted as the straight line space of the spherical model of the Möbius plane. See the möbiusebene-book-chapter

Möbius - Geradenraum for this transfer principle. The Lie product [ , ] is defined like in the Euclidean vector space the cross product

by the uniquely determined linear form for all

is thus nothing else than a complexification of the Euclidean vector space. The applet above is a real simplification of the complex relations: e.g. it cannot be represented that every STRAIGHT line in the complex plane intersects the quadric

at one or two points: in

every quadratic equation is solvable! The POINTS on the Möbius quadric

with exception of

are achieved through the complex parametrisation:

There is thus a 1 to 1 relationship between the Möbius points in

and the POINTS on

. The group of orientation-preserving Möbius transformations turns out to be isomophic to

. Even more:

is the LIE algebra of this group! We call the vectors used in the tables a Euclidean coordinate system of

Brief interpretation of the base vectors: The

-plane

is projected stereographically onto the unit sphere.

is a tangent to the unit sphere in the direction of the

-axis,

is a tangent in

, also in

-direction.

is the straight line connecting these two points, i.e. the

-axis in space.

is a tangent to the sphere at the image point of the stereographic projection of

is the connecting line of the stereographic images of

and

This representation of plane Möbius geometry has disadvantages, but also very many advantages: The circles as individual objects are not easily accessible! In contrast, there is the variety of possible interpretations of the POINTS and the vectors of

The - projectively to be seen - POINTS on - i.e. it is - are the points of Möbus geometry.
The vectors with can be interpreted as tangential vectors: if is a differentiable curve, then is tangential to the curve. can be real or complex. In the 2nd case, complex-analytical functions are captured!
The vectors can be interpreted as infinitesimal Möbius movements: the linear mappings , explained by for all , act on the Möbius points on . The trajectories of the motions are, depending on the type of vector for real parameters t hyperbolic (), or elliptic () or parabolic () pencils of circles; for one obtains loxodromic trajectory curves, these are the curves which intersect a hyperbolic ( - - or an elliptical - - ) pencil of circles at a constant angle.
The movements are one-parameter subgroups of the Möbius group. Such movements of a group are called w-movements. Here, too, one obtains a real - - or a complex - - subgroup.

Ein lineares Vektorfeld

The tangential vectors of the trajectories of a w-motion on the quadric generate a linear vector field: , with . See the book chapter Kreisbüschel oder lineare Vektorfelder
The vectors with can be interpreted as straight line vectors in the sphere model of the Möbius plane. The STRAIGHT line with intersects the sphere at 2 points. The STRAIGHT line is the non-intersecting polar to it!
The quadratic vector fields are also of interest: with . The calculation yields an elliptic differential equation , hose solution curves for special positions of the focal points are confokal bicircular quartics; this is the case, for example, if the focal points lie on a circle!
If you let the "focal points" at the top or bottom of the applet run against each other, the pencil of circles and the trajectories approach the circles of a parabolic pencil of circles !

Question: What are the trajectories of w-movements in the group of LORENTZ transformations? Since

is isomorphic to the group of orthochronous orientation-preserving LORENTZ transformations,

is isomorphic to the LIE-Algebra of this group!

This vector field is constructed with the formulas of the transmission principle given above: To

are calculated

. The connecting line in the spherical model is

. The direction vector

in the point

is calculated using the linear vector field

. Thanks gegebra, all complex calculations are problem-free!

the möbiusgroup SO(3, ℂ)

Ein lineares Vektorfeld

Lösungskurven eines linearen Vektorfeldes

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