Der methodische Einsatz von MÖBIUS-Transformationen über den Quaternionen in der Geometrie des Raumes

  • The methodic application of Möbius transformations over the quaternions in geometry of three dimensional space

Hermanns, Wencke; Krieg, Aloys (Thesis advisor)

Aachen : Publikationsserver der RWTH Aachen University (2007)
Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2007


Studies in plane geometry can benefit from identifying $R^2$ with $C$ and exploiting analytic properties of the complex numbers. This motivates the investigation, whether geometry in three dimensional space can profit in a similar way from applying of the analytical properties of the quaternions when $R^3$ is identified with the imaginary space $Imh$ of the quaternions. Hence this paper starts with a general basic introduction to the properties of the skewfield $h$ of Hamiltonian quaternions and with the extension of $h$ by the point $infty$ to the compactification $widehat{h}$ by stereographic projection. In particular we will find that two quaternions are conjugate to each other if and only if their real parts and absolute values coincide. Using the characterization of conjugation and the skewfield properties of $h$ it is possible to simplify a quaternionic polynomial equation of degree 2 of the form $XcX+Xd-aX-b=0$, with $a,b,c,dinh$, $ceq 0$ and $b-ac^{-1}deq 0$, where $X$ is the variable, and to characterise the solvability of this equation in $h$. The existence of a solution of this equation is derived using elementary mathematical methods. In some cases a statement about shape and number of solutions is made. In the following part fundamentals of geometry in $widehat{h}$ and $Imwidehat{h}$ are discussed. Among other things the circumcenter of three points from $Imh$ is specified as well as the circumcenter and circumradius of a tetrahedron in $Imwidehat{h}$. The representations of these points are required for the generalisations of the tetrahedron and parallel version of Roberts' Theorem by continuity arguments. For this also the introduction of the power of a point with respect to a sphere is necessary as well as the terms radical plane, line and point. In the next chapter quaternionic Möbius transformations are introduced and their properties characterised. For example the group $M$ of the Möbius transformations is isomorphic to the projective general linear group $mathrm{PGL}_2(h)=GL(2;h)/left(R^starcdot Eight)$ and these transformations map 3-spheres and hyperplanes onto 3-spheres or hyperplanes. After that those transformations which map $Imwidehat{h}$ onto $Imwidehat{h}$ are determined. They are found to be given by ${MinGL(2; h);, phi_M(Imwh)=Imwh}=R^starcdotP$ with $P = {Minmat(2;h);, MQoverline{M}^t=Q}dot{cup} {Minmat(2;h);, MQoverline{M}^t=-Q}= P_+dot{cup} , P_-$ and $Q=egin{pmatrix} 0 & 1\ 1 & 0 end{pmatrix}in GL (2;h)$. Especially those transformaions are determined which map the 2-sphere $S^2$ to itself. Then the focus is on the fixed point structure and the iterative behaviour of the Möbius transformations which map $Imwidehat{h}$ onto $Imwidehat{h}$. It is found that these Möbius transformations have fixed points in $Imwidehat{h}$ only under certain conditions. Because both the fixed point structure and iterative behaviour are invariant under conjugation the fixed point structure and iterative behaviour of the Möbius transformations from $Imwidehat{h}$ to $Imwidehat{h}$ are characterised by specifying simple elements of the conjugacy class and examining their fixed point structure and iterative behaviour. June Lester has developed the theory of cross ratios in the compactified complex plane in several publications. She proves many identities for the cross ratio and focuses on its geometric properties. A similar aim is now strived for: The cross ratio of four distinct points in $widehat{h}$ is defined and elementary properties of this cross ratio are derived. Because $h$ is a skewfield, i.e. the elements do not commute with respect to multiplication, the cross ratio of four distinct points in $widehat{h}$ can not be defined as in Lester's case but one has to switch to conjugacy classes. Amongst other things, relationships are made between Möbius transformations and cross ratios: For example it is proven that cross ratios are invariant under Möbius transformations and that there exists a Möbius transformations which maps $q_n$ to $w_n$ for $1leq n leq 4$, where $q_1,q_2,q_3,q_4inwidehat{h}$ and $w_1,w_2,w_3,w_4inwidehat{h}$ are four mutually distinct points, respectively, if and only if the cross rations $[q_1,q_2,q_3,q_4]$ and $[w_1,w_2,w_3,w_4]$ are equal. Moreover, geometrical properties of the cross ratios are considered. For example it is possible to characterise if four distinct points in $widehat{h}$ lie on a circle or a straight line and if three not collinear points build a equilateral triangle. In the following chapter the triangle and parallel versions of Miquel's Theorem are proven for any plane in the compactified imaginary space $Imwidehat{h}$. In addition those theorems are transferred to conditions of intersection on a 2-sphere. These different versions of Miquel's Theorem are necessary in order to prove of Roberts' Theorem. In the last chapter the tetrahedron and parallel versions of Roberts' Theorem are proved following Nathan Altshiller-Court. Using arguments of continuity leads to general versions of the theorems. Moreover, Möbius transformations are used upon the tetrahedron and parallel versions of Roberts' Theorem whereby new theorems of intesection are found. In addition to these examples there are many other possibilities to get new theorems of intersection using Möbius transformations. So these transformations present a flexible instrument for geometry in three dimensional space.