[br][table][tr][td][url=https://www.geogebra.org/m/nzfg796n#material/rkhmqpb9][img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACUAAAA2CAYAAABA3FA2AAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsQAAA7EAZUrDhsAAACpSURBVGhD7dkxCsJAFEXR/wZiJWJhIW7MUnApriwLEFdhZy0iiN8M2tjdLr94h8wEUt3yQaRhyMiMiH7mpulRv0vU/Gm/dymOohxFOYpyFOUoylGUoygdd9tye0qv1upFTUVenoSjKEdRjqIcRTmKchTlKErX/abgyLvUG3nKs5cn4ijKUZSjKEdRjqIcRRWN6j9six2dxkMu9YiV7t+Ps8l45iJu73V8AE/fHKUjFbbZAAAAAElFTkSuQmCC[/img][/url][/td][td][size=50] this activity is a page of [color=#980000][i][b]geogebra-book[/b][/i][/color][br] [url=https://www.geogebra.org/m/y9cj4aqt][color=#0000ff][u][i][b]elliptic functions & bicircular quartics & . . .[/b][/i][/u][/color][/url]([color=#ff7700][i][b]27.04.2023[/b][/i][/color])[/size][/td][/tr][/table][right][size=50] [b][i]this activity is also a page of[/i][/b][/size][size=50][i][b] [color=#980000][i][b]GeoGebrabook[/b][/i][/color] [url=https://www.geogebra.org/m/kCxvMbHb]Moebiusebene[/url] ([color=#ff7700]29.09.2020[/color][/b][/i])[/size][/right]

[size=85]In the complex, 3-dimensional vector space[/size] [math]\large\mathcal{G}[/math] [size=85]with non-degenerate quadratic form [math] \bullet [/math][br]an [i]oriented[/i] basis is selected[/size] [math]\mathbf\vec{p}_\infty,\,\mathbf\vec{g}_0,\,\mathbf\vec{p}_0[/math] [size=85]with[/size] [math]\mathbf{Det}\left(\mathbf\vec{p}_\infty,\mathbf\vec{g}_0, \mathbf\vec{p}_0\right)=1[/math][size=85], [br]for which the two [b][i]product tables[/i][/b] are to apply:[/size][br][list][br] [math]\Large\begin{tabular} {|c||c|c|c|} \hline \bullet & \mathbf\vec{p}_\infty & \mathbf\vec{g}_0 & \mathbf\vec{p}_0 \\ \hline\hline \mathbf\vec{p}_\infty & 0 & 0 & 1 \\ \hline \mathbf\vec{g}_0 & 0 & -1 & 0 \\ \hline \mathbf\vec{p}_0 & 1 & 0 & 0 \\ \hline \end{tabular}[/math] [math]\Large\begin{tabular} {|c||c|c|c|} \hline [\;\,,\;] & \mathbf\vec{p}_\infty & \mathbf\vec{g}_0 & \mathbf\vec{p}_0 \\ \hline\hline \mathbf\vec{p}_\infty & \mathfrak{o} & \mathbf\vec{p}_\infty & \mathbf\vec{g}_0 \\ \hline \mathbf\vec{g}_0 & - \mathbf\vec{p}_\infty & \mathfrak{o} & \mathbf\vec{p}_0 \\ \hline \mathbf\vec{p}_0 & - \mathbf\vec{g}_0 & - \mathbf\vec{p}_0& \mathfrak{o} \\ \hline \end{tabular}[/math][br][/list][br][size=85]The name [math]\large\mathcal{G}[/math] is chosen because this vector space can be interpreted as the[i][b] straight line space[/b][/i] of [br]the spherical model of the [b][i][color=#0000ff]Möbius plane.[/color][/i][/b][br]See the [color=#980000][i][b]möbiusebene-book[/b][/i][/color]-chapter [math]\hookrightarrow[/math] [u][color=#0000ff][i][b][url=https://www.geogebra.org/m/kCxvMbHb#chapter/168840]Möbius - Geradenraum[/url] [/b][/i][/color][/u] for this transfer principle.[br][br]The [b]Lie[/b] product[/size][size=85] [ , ] is defined like in the [b][i][color=#ff00ff]Euclidean vector space[/color][/i][/b] the [b][i]cross product[/i][/b] [math]\otimes[/math]:[br][/size][list][*][size=85][math]\left(\mathbf\vec{g}_1,\,\, \mathbf\vec{g}_2\right)\; \mapsto\;\left[\,\mathbf\vec{g} _{1}\,,\,\mathbf\vec{g}_2\,\right][/math][/size] [size=85]by the uniquely determined linear form [/size] [math]\mathbf{Det}\left(\mathbf\vec{g},\mathbf\vec{g}_1, \mathbf\vec{g}_2\right)=\mathbf\vec{g}\bullet \left[\,\mathbf\vec{g} _{1}\,,\,\mathbf\vec{g}_2\,\right][/math] [size=50][size=85] for all [/size][/size][math]\mathbf\vec{g}\in \large\mathcal{G}[/math][/*][/list][math]\left(\large\mathcal{G}\;,\bullet,\;\left[\;,\;\right]\;\right)[/math] [size=85]is thus nothing else than a complexification of the [/size][size=85][b][i][color=#ff00ff]Euclidean vector space[/color][/i][/b][/size][size=85].[/size][size=85][br]The applet above is a real simplification of the [b][i]complex[/i][/b] relations: [br]e.g. it cannot be represented that every STRAIGHT line in the complex plane intersects the quadric [math]\large\mathcal{Q}[/math] [br]at one or two points: in [math]\mathbb{C}[/math] every quadratic equation is solvable![br] [/size][size=85][size=85]The POINTS on the Möbius quadric[/size][/size][size=85][size=85] [math]\large\mathcal{Q}[/math] with exception of [/size] [math]\infty\equiv \mathbf\vec{p}_\infty[/math] [/size][size=85][size=85]are achieved through the complex parametrisation:[/size][/size][br][list] [math]\mathbf\vec{p}(z):=\frac{z^2}{2}\cdot \mathbf\vec{p} _\infty+z\cdot\mathbf\vec{g}_0+\mathbf\vec{p}_0,\mbox{ with }z\in\mathbb{C}[/math] [/list][size=85]There is thus a 1 to 1 relationship between the [b][i][color=#0000ff]Möbius points[/color][/i][/b] in[/size][size=85] [math]\mathbb{C}\cup \{ \infty \}[/math] [/size][size=85]and the POINTS on[/size][size=85] [math]\large\mathcal{Q}[/math].[br][/size][size=85]The group of [/size][size=85][i]orientation-preserving[/i][/size][size=85] [b][i][color=#0000ff]Möbius transformations[/color][/i][/b] turns out to be isomophic to[/size][size=85] [math]\mathbf{SO\left(3,\mathbb{C}\right)}[/math].[br][/size][size=85]Even more:[/size][size=85] [math]\left(\large\mathcal{G}\;,\bullet,\;\left[\;,\;\right]\;\right)[/math] [/size][size=85]is the [b]LIE[/b] algebra of this group![/size][size=85] [br]We call the vectors used in the tables a [b][i][color=#ff00ff]Euclidean coordinate system[/color][/i][/b] of[/size][size=85] [math]\large\mathcal{G}[/math].[/size]

[size=50][b][i][u][color=#cc0000]Brief interpretation of the base vectors:[/color][/u][/i][/b][br]The [math]xy[/math]-plane [math]\mathbb{C}\cup\left\{\infty\right\}[/math] is projected [b][i][color=#0000ff]stereographically[/color][/i][/b] onto the [b][i][color=#bf9000]unit sphere[/color][/i][/b].[br] [math]\mathbf\vec{p}_0[/math]is a [b][i][color=#666666]tangent[/color][/i][/b] to the unit sphere in the direction of the [math]x[/math]-axis,[br] [math]\mathbf\vec{p}_\infty[/math] is a [/size][size=50][b][i][color=#666666]tangent[/color][/i][/b][/size][size=50] in [math]\infty[/math], also in [math]x[/math]-direction. [br][math]\mathbf\vec{g}_0[/math] is the straight line connecting these two [b][i][color=#ff0000]points[/color][/i][/b], i.e. the [math]z[/math]-axis in space.[br][math]\mathbf\vec{p}(z)[/math] is a [/size][size=50][b][i][color=#666666]tangent[/color][/i][/b][/size][size=50] to the [b][i][color=#980000]sphere[/color][/i][/b] at the image [b][i][color=#ff0000]point[/color][/i][/b] of the [b][i][color=#0000ff]stereographic[/color][/i][/b] projection of [math]z[/math].[br][math]\mathbf{\vec{g}}=\frac{1}{\mathbf{\vec{p}\left(z_1\right)}\bullet \mathbf{\vec{p}\left(z_2\right)}}\cdot\left[\mathbf{\vec{p}\left(z_1\right)},\mathbf{\vec{p}\left(z_2\right)}\right][/math] is the [b][i][color=#666666]connecting line[/color][/i][/b] of the [b][i][color=#0000ff]stereographic[/color][/i][/b] images of [math]z_1[/math] and [math]z_2[/math].[br][/size]

[size=85]This representation of [b][i][color=#0000ff]plane Möbius geometry[/color][/i][/b] has disadvantages, but also very many advantages:[br]The [b][i][color=#ff0000]circles[/color][/i][/b] as individual objects are not easily accessible![br]In contrast, there is the variety of possible interpretations of the [b][color=#ff0000]POINTS[/color][/b] and the vectors of [math]\large\mathcal{G}[/math].[br][/size][list][*][size=85]The - [math]\hookrightarrow[/math] [url=https://www.geogebra.org/m/kCxvMbHb#material/fu38rxw6]projectively to be seen[/url] - POINTS [math] \left[\;\mathbf\vec{p}\;\right][/math] on [math]\large\mathcal{Q}[/math]- i.e. it is [math]\mathbf\vec{p}\bullet \mathbf\vec{p}=0 [/math] - are the [b][i][color=#ff0000]points[/color][/i][/b] of [b][i][color=#0000ff]Möbus geometry[/color][/i][/b].[/size][br][/*][*][size=85]The vectors [math] \mathbf\vec{p}[/math] with [math]\mathbf\vec{p}\bullet \mathbf\vec{p}=0 [/math] can be interpreted as [b][i]tangential vectors[/i][/b]: [br]if [math]z(t)[/math] is a differentiable curve, then [math]\mathbf\vec{p}(t):=\frac{1}{z'(t)}\cdot\left(\frac{z^2}{2}\cdot \mathbf\vec{p} _\infty+z\cdot\mathbf\vec{g}_0+\mathbf\vec{p}_0\right)[/math] is tangential to the curve. [br][math] t[/math] can be real or complex. In the 2nd case, [b][i][color=#38761d]complex-analytical functions[/color][/i][/b] are captured![/size][br][/*][*][size=85]The vectors [math] \mathbf\vec{g}\in \large\mathcal{G}[/math] can be interpreted as [b][i][color=#0000ff]infinitesimal Möbius movements[/color][/i][/b]:[br]the linear mappings [math] \mathbf\vec{g} \mapsto \mathbf{ad}\; \mathbf\vec{g}[/math], explained by [math] \mathbf{ad}\; \mathbf\vec{g} \left(\mathbf\vec{\tilde{g}}\right) = \left[ \mathbf\vec{g}, \mathbf\vec{\tilde{g}}\right][/math] for all [math]\mathbf\vec{\tilde{g}} \in \large\mathcal{G} [/math] , [br]act on the [b][i][color=#0000ff]Möbius[/color][/i][/b] [b][i][color=#ff0000]points[/color][/i][/b] on [math]\large\mathcal{Q}[/math]. [br]The trajectories of the motions [math]t\mapsto\mathbf{exp}(t\cdot \mathbf{ad}\,\mathbf\vec{g})[/math] are, depending on the type of vector [math]\mathbf\vec{g}[/math] for real parameters t [br] [/size][size=85][size=85][i][b]hyperbolic[/b][/i][/size] ([math]\mathbf\vec{g}\bullet\mathbf\vec{g} >0[/math]), or [/size][size=85][size=85][i][b]elliptic [/b][/i][/size] ([math]0>\mathbf\vec{g}\bullet\mathbf\vec{g} [/math]) or [i][b]parabolic[/b][/i] ([math]\mathbf\vec{g}\,^2=0 [/math]) [b][i][color=#ff0000]pencils of circles[/color][/i][/b]; [br]for [math]\mathbf\vec{g}\,^2\ne 0, \mathbf\vec{g}\,^2\notin\mathbb{R}[/math] one obtains [b][i][color=#0000ff]loxodromic trajectory curves[/color][/i][/b], [br]these are the curves which intersect a [b][i][color=#ff0000]hyperbolic[/color][/i][/b] ( - - or an [b][i][color=#ff0000]elliptical [/color][/i][/b]- - )[color=#ff0000][i][b] pencil of circles[/b][/i][/color] at a constant angle. [/size][/*][*][size=85]The movements [math]t\mapsto\mathbf{exp}(t\cdot \mathbf{ad}\,\mathbf\vec{g})[/math] are [b][i]one-parameter subgroups[/i][/b] of the [b][color=#0000ff][i]Möbius group[/i][/color][/b]. [br]Such movements of a group are called [b][i][color=#0000ff]w-movements[/color][/i][/b]. [br]Here, too, one obtains a [i]real[/i] - [math]t\in\mathbb{R}[/math] - or a [i]complex[/i] - [math]t\in\mathbb{C}[/math] - subgroup.[br][/size][/*][/list]

[size=85][br][list][*]The [b][i][color=#00ffff]tangential vectors[/color][/i][/b] of the trajectories of a [b][i]w-motion[/i][/b] [math]t\mapsto\mathbf{exp}(t\cdot \mathbf{ad}\,\mathbf\vec{g})[/math] on the [b][i][color=#ff7700]quadric[/color][/i][/b] [math]\mathbf{\mathcal{Q}}[/math] generate [br]a linear vector field: [math]\mathbf\vec{g}\bullet\mathbf\vec{p}=1[/math], with [math]\mathbf\vec{p}\bullet\mathbf\vec{p}=0[/math]. [size=50]See the [b][color=#980000]book[/color][/b] chapter [math]\hookrightarrow[/math][/size] [color=#0000ff][u][i][b][url=https://www.geogebra.org/m/kCxvMbHb#chapter/168949]Kreisbüschel oder lineare Vektorfelder[/url][/b][/i][/u][/color][br][br][/*][*]The vectors [math]\mathbf\vec{g}\in\mathbf{\mathcal{G}}[/math] with [math]\mathbf\vec{g}\,^2\in \mathbb{R}[/math] can be interpreted as [b][i][color=#ff0000]straight line vectors[/color][/i][/b] in the [b][i][color=#980000]sphere[/color][/i][/b] model of the[color=#0000ff] [/color][b][i][color=#0000ff]Möbius plane[/color][/i][/b]. [br]The STRAIGHT line [math]\mathbf\vec{g}\in\mathbf{\mathcal{G}}[/math] with [math]\mathbf\vec{g}\,^2 < 0 [/math] intersects the [size=85][b][i][color=#980000]sphere[/color][/i][/b][/size] at [b][color=#cc0000]2[/color][/b] [b][i][color=#ff0000]points[/color][/i][/b].[br]The STRAIGHT line [math]i\cdot \mathbf\vec{g}[/math] is the non-intersecting [b][i]polar[/i][/b] to it![/*][*]The [b][i]quadratic vector fields[/i][/b] are also of interest: [math]\mathbf\vec{g}_1\bullet\mathbf\vec{p}\cdot\mathbf\vec{g}_2\bullet\mathbf\vec{p} =1[/math] with [math]\mathbf\vec{p}\bullet\mathbf\vec{p}=0[/math]. [br]The calculation yields an [b][i][color=#9900ff]elliptic differential equation[/color][/i][/b] [math]\left(z'\right)^2=c\cdot\left(z-f_1\right)\cdot\left(z-f_2\right)\cdot\left(z-f_3\right)\cdot\left(z-f_4\right)[/math], [br]hose solution curves for special positions of the f[b][i][color=#00ff00]ocal points[/color][/i][/b] [math] f_1, f_2, f_3, f_4 [/math][color=#ff7700][i][b] [/b][/i][/color]are[color=#ff7700][i][b][br][/b][/i][/color][color=#38761d][i][b]confokal[/b][/i][/color][color=#ff7700][i][b] bicircular quartics[/b][/i][/color]; this is the case, for example, if the [b][i][color=#00ff00]focal points[/color][/i][/b] lie on a [b][i][color=#ff0000]circle[/color][/i][/b]![/*][*]If you let the "[b][i][color=#00ff00]focal points[/color][/i][/b]" at the top or bottom of the applet [math]z_1,z_2[/math] run against each other, [br]the [b][i][color=#ff0000]pencil of circle[/color][/i][/b]s and the [b][i][color=#cc0000]trajectories[/color][/i][/b] approach the [b][i][color=#ff0000]circles[/color][/i][/b] of a [b][i][color=#ff0000]parabolic pencil of circles[/color][/i][/b] ![br][/*][/list][u][i][b][color=#cc0000]Question:[/color][/b][/i][/u] What are the[b][i] trajectories[/i][/b] of [b][i][color=#0000ff]w-movements[/color][/i][/b] in the group of [b]LORENTZ[/b] [b][i][color=#0000ff]transformations[/color][/i][/b]? [br]Since [math]\mathbf{SO\left(3,\mathbb{C}\right)}[/math] is [b][i]isomorphic[/i][/b] to the group of [b][i][color=#0000ff]orthochronous[/color][/i][/b] [b][i][color=#38761d]orientation-preserving[/color][/i][/b] [b]LORENTZ[/b] [b][i][color=#0000ff]transformations[/color][/i][/b], [/size][size=85][br] [math]\left(\large\mathcal{G}\;,\bullet,\;\left[\;,\;\right]\;\right)[/math] [/size][size=85]is [b][i]isomorphic[/i][/b][/size][size=85] to the [b]LIE[/b]-Algebra of this group![/size]

[size=85]This vector field is constructed with the formulas of the[b][i] transmission principle[/i][/b] given above:[/size][size=85][br]To [math]z_1,z_2,z\in\mathbb{C}[/math] [/size][size=85]are calculated[/size][size=85] [math]\mathbf{\vec{p}(z_1)},\mathbf{\vec{p}(z_2)},\mathbf{\vec{p}(z)}[/math]. [br][/size][size=85]The [b][i][color=#0000ff]connecting line[/color][/i][/b] in the spherical model is[/size][size=85] [math]\mathbf{\vec{g}}=\frac{1}{\mathbf{\vec{p}\left(z_1\right)}\bullet \mathbf{\vec{p}\left(z_2\right)}}\cdot\left[\mathbf{\vec{p}\left(z_1\right)},\mathbf{\vec{p}\left(z_2\right)}\right][/math].[br][/size][size=85]The direction vector [/size][size=85] [math]w[/math] in the point [math]z[/math] [/size][size=85]is calculated using the [b][i][color=#134f5c]linear vector field[/color][/i][/b] [/size][size=85] [math]\left(e^{\varphi\cdot i}\cdot \mathbf{\vec{g}\right)\bullet \mathbf{\vec{p}(z)}=w[/math].[br]Thanks [color=#980000][i][b]ge[/b][/i][/color][icon]/images/ggb/toolbar/mode_circle2.png[/icon][color=#980000][i][b]gebra[/b][/i][/color], all complex calculations are problem-free![/size]