[size=85]Four (different) points [b]*)[/b] [/size][math]\mathbf\vec{p}_i=\mathbf\vec{p}\left(z_i\right)\in\mathbf\mathcal{ G}, i=1,...,4[/math] [size=85]can be connected in three ways [br]in pairs by straight lines: ( [b]*)[/b] see the [math]\hookrightarrow[/math] [b][u][color=#0000ff][url=https://www.geogebra.org/m/y9cj4aqt#material/dwwkcd3z]activity before[/url][/color][/u][/b] )[/size][list][*][math]\mathbf\vec{g}_{12}=\left[\,\mathbf\vec{p}_1\,,\,\mathbf\vec{p}_2\,\right] [/math] [size=85]and[/size] [math]\mathbf\vec{g}_{34}=\left[\,\mathbf\vec{p}_3\,,\,\mathbf\vec{p}_4\,\right] [/math][/*][size=85][/size][size=85][/size][size=50][/size][br][*][math]\mathbf\vec{g}_{13}=\left[\,\mathbf\vec{p}_1\,,\,\mathbf\vec{p}_3\,\right] [/math] [size=85]and[/size] [math]\mathbf\vec{g}_{24}=\left[\,\mathbf\vec{p}_2\,,\,\mathbf\vec{p}_4\,\right] [/math] [/*][size=50][/size][br][*][math]\mathbf\vec{g}_{14}=\left[\,\mathbf\vec{p}_1\,,\,\mathbf\vec{p}_4\,\right] [/math] [size=85]and[/size] [math]\mathbf\vec{g}_{23}=\left[\,\mathbf\vec{p}_2\,,\,\mathbf\vec{p}_3\,\right] [/math] [br][/*][/list][br][size=85]For the [b]LIE[/b]-products[/size] [math]\mathbf\vec{g}_{1234}=\left[\,\mathbf\vec{g}_{12}\,,\,\mathbf\vec{g}_{34}\,\right] [/math] , [math]\mathbf\vec{g}_{1324}=\left[\,\mathbf\vec{g}_{13}\,,\,\mathbf\vec{g}_{24}\,\right] [/math] [size=85]and[/size] [math]\mathbf\vec{g}_{1423}=\left[\,\mathbf\vec{g}_{14}\,,\,\mathbf\vec{g}_{23}\,\right] [/math] [size=85]can easily be calculated [br]as follows (with the help of the [b][i]general development rule[/i][/b], calculation example see below [b]**)[/b]):[/size][br] [math]\mathbf\vec{g}_{1234}\bullet\mathbf\vec{g}_{1324}=\mathbf\vec{g}_{1324}\bullet\mathbf\vec{g}_{1423}=\mathbf\vec{g}_{1423}\bullet\mathbf\vec{g}_{1234} = 0 [/math] [br][size=85]If one thinks of the connecting line vectors as normalised from the outset: [/size][math]\mathbf\dot{\vec{g}}_{ij}=\frac{\left[\,\mathbf\vec{p}_i\,,\,\mathbf\vec{p}_j\,\right]}{\mathbf\vec{p}_i\cdot \mathbf\vec{p}_j}\ , i \ne j ,[/math] [size=85]the following apply[/size][br] [math]\mathbf\dot{\vec{g}}_{ij}\bullet \mathbf\dot{\vec{g}}_{ij} = \frac{\mathbf\vec{p}_i\bullet \mathbf\vec{p}_i*\mathbf\vec{p}_j\bullet \mathbf\vec{p}_j-\left(\mathbf\vec{p}_i\bullet \mathbf\vec{p}_j\right)^2}{\left(\mathbf\vec{p}_i\bullet \mathbf\vec{p}_j\right)^2} = -1\ , i \ne j ,[/math] [br][size=85]and for the [b]LIE[/b]-products [/size][math]\mathbf\dot{\vec{g}}_{1234}=\left[\,\mathbf\dot{\vec{g}}_{12}\,,\,\mathbf\dot{\vec{g}}_{34}\,\right] [/math] , [math]\mathbf\dot{\vec{g}}_{1324} [/math] [size=85]and[/size] [math]\mathbf\dot{\vec{g}}_{1423} [/math][size=85] (defined accordingly) is thus obtained:[br] [/size][math]\left(\mathbf\dot{\vec{g}}_{1234}\right)^2=\left(\mathbf\dot{\vec{g}}_{1324} \right)^2=\left(\mathbf\dot{\vec{g}}_{1423} \right)^2=1[/math]. [br][size=85]These straight line vectors therefore form in [/size][math]\large\mathbf\mathcal{ G}[/math] [size=85]an [b]ON-Basis[/b].[/size] [size=85]The poles of these basis vectors are the intersections [br]of 3 pairwise [b][i][color=#0000ff]orthogonal[/color][/i][/b] [b][i][color=#ff0000]circles[/color][/i][/b] and can, with suitable orientation, be mapped by a [b][i][color=#0000ff]Möbius transformation[/color][/i][/b] [br]to the point pairs [math]\{0\,,\,\large\infty\}[/math] , [math]\{ -1\,,\, 1\}[/math] and [math]\{ -i\,,\, i \}[/math] . [br]Becaue of [math]\mathbf\vec{g}_{1234}=\left[\,\mathbf\vec{g}_{12}\,,\,\mathbf\vec{g}_{34}\,\right] [/math] follows [math]\mathbf\vec{g}_{1234}\bullet\mathbf\vec{g}_{13}=\mathbf\vec{g}_{1234}\bullet\mathbf\vec{g}_{34}= 0 [/math], thus the poles [math]z12_1[/math] and [math]z12_2[/math] from [math]\mathbf\vec{g}_{1234} [/math] [br]separate the point pairs [math]\{\,z_1\, ,\, z_2\,\}[/math] and [math]\{\,z_3\, ,\,z_4\,\}[/math] harmonically. [br] Be[/size][size=85] [math]\{\,0\,, \,\large\infty\,\}[/math] the poles of [math]\mathbf\vec{g}_{1234} [/math] after the [/size][size=85][b][i][color=#0000ff]Möbius transformation[/color][/i][/b][/size][size=85], then [br] - the images [math]b_i[/math] of [math]\{\,z_1\, ,\, z_2\,\}[/math] resp. [math]\{\,z_3\, , \,z_4\,\}[/math] [size=85]are point-symmetrical to the origin: [math]b_2=-b_1[/math] and [math]b_4=-b_3[/math] . [br] - The images of the point pairs [math]\{\,z_1\, , \,z_3\,\}[/math] resp.. [math]\{\,z_2\, ,\, z_4\,\}[/math] are correspondingly harmonic to [math]\{\, -1\,, \,1\, \}[/math], [br] i.e. [math]b_3=\frac{1}{b_1}[/math] and [math]b_4=\frac{1}{b_2}[/math] must apply.[br][u][i][b]Conclusion:[/b][/i][/u] 4 different points of the [b][i][color=#0000ff]Möbius plane[/color][/i][/b] can always be mapped by a [/size][/size][size=85][b][i][color=#0000ff]Möbius transformation[/color][/i][/b][/size][size=85][size=85] [br] to [math]\{\,f\,,\,-f\,,\,\frac{1}{f}\,,\,-\frac{1}{f}\,\}[/math], for a suitable one [math]f\in\mathbb{C}[/math]. [br] We call this the representation of the 4 points in [b]normal form[/b].[br]. [br][/size][br][/size][size=50][u][i][b]Note on the underlying calculations for the applet above:[/b][/i][/u] [br]the complex numbers [/size][math]z_i[/math] [size=50]are mapped in the [b][color=#ff00ff]Euclidean KOS[/color][/b] to the complex touch line vectors[/size][math]\small\mathbf\vec{p}_i=\mathbf\vec{p}\left(z_i\right), i=1,...,4[/math][size=50]. [br][b]LIE [/b]products are calculated with the complex cross product. [br]To calculate the poles of a straight line vector [/size][math]\small\mathbf\vec{g}\in\mathbf\mathcal{ G}[/math][size=50] one must determine the [b][i][color=#ff00ff]solutions [/color][/i][/b]of the [b][i]complex quadratic equation[/i][/b][/size]:[math]\small\mathbf\vec{g}\bullet\mathbf\vec{p}\left(z\right)=0[/math][size=50].[/size][br][br][size=85][b]**)[/b] Reason for the equation [/size] [math]\mathbf\vec{g}_{1234}\bullet\mathbf\vec{g}_{1324} = 0 [/math] [br][br] [math]\mathbf\left[\left[\,\mathbf\vec{p}_1\,,\,\mathbf\vec{p}_2\,\right]\,,\,\left[\,\mathbf\vec{p}_3\,,\,\mathbf\vec{p}_4\,\right]\,\right]\bullet \mathbf\left[\left[\,\mathbf\vec{p}_1\,,\,\mathbf\vec{p}_3\,\right]\,,\,\left[\,\mathbf\vec{p}_2\,,\,\mathbf\vec{p}_4\,\right]\,\right] = [/math] [br] [math]= \left(\left[\,\mathbf\vec{p}_1\,,\,\mathbf\vec{p}_2\,\right]\bullet \left[\,\mathbf\vec{p}_1\,,\,\mathbf\vec{p}_3\,\right]\right) \cdot \left(\left[\,\mathbf\vec{p}_3\,,\,\mathbf\vec{p}_4\,\right]\bullet \left[\,\mathbf\vec{p}_2\,,\,\mathbf\vec{p}_4\,\right]\right) - \left(\left[\,\mathbf\vec{p}_3\,,\,\mathbf\vec{p}_4\,\right]\bullet \left[\,\mathbf\vec{p}_1\,,\,\mathbf\vec{p}_3\,\right]\right) \cdot \left(\left[\,\mathbf\vec{p}_1\,,\,\mathbf\vec{p}_2\,\right]\bullet \left[\,\mathbf\vec{p}_2\,,\,\mathbf\vec{p}_4\,\right]\right)[/math] [br] [math]= \left(-\left(\,\mathbf\vec{p}_1\bullet\mathbf\vec{p}_2\,\right)\cdot \left(\,\mathbf\vec{p}_1\bullet\mathbf\vec{p}_3\,\right)\right) \cdot \left(-\left(\,\mathbf\vec{p}_3\bullet\mathbf\vec{p}_4\,\right)\cdot\left(\,\mathbf\vec{p}_2\bullet\mathbf\vec{p}_4\,\right)\right) - \left(\left(\,\mathbf\vec{p}_3\bullet\mathbf\vec{p}_4\,\right)\cdot \left(\,\mathbf\vec{p}_1\bullet\mathbf\vec{p}_3\,\right)\right) \cdot \left(\left(\,\mathbf\vec{p}_1\bullet\mathbf\vec{p}_2\,\right)\cdot \left(\,\mathbf\vec{p}_2\bullet\mathbf\vec{p}_4\,\right)\right) = 0[/math] [br]