[table][tr][td][url=https://www.geogebra.org/m/nzfg796n#material/ehxdqrrk[br]][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=#0000ff][u][i][b][/b][/i][/u][/color]([color=#ff7700][i][b]06.02.2023[/b][/i][/color])[/size][/td][/tr][/table][center][size=100][b][color=#0000ff]move[/color] [color=#ff0000]z[sub]1[/sub][/color], [color=#ff0000]z[sub]2[/sub][/color], [color=#ff00ff]z[sub]3[/sub][/color], [color=#ff00ff]z[sub]4[/sub][/color][/b][/size][/center]

[size=85]To [b][color=#cc0000]4[/color][/b] different [b][i][color=#ff0000]points[/color][/i][/b] [math]z_1,z_2,z_3,z_4[/math] in [math]\mathbb{C}\cup\left\{\infty\right\}[/math] there is a [b][i][color=#0000ff]Möbius transformation [/color][/i][/b]ie.: a rational function of the form[br][/size][list][*][size=85] [math]z\mapsto T\left(z\right)=\frac{a\cdot z+b}{c\cdot z+d}[/math],[/size][/*][/list][size=85]which maps the [b][i][color=#ff0000]points[/color][/i][/b] to [b][color=#cc0000]4[/color][/b] complex image [b][i][color=#ff0000]points[/color][/i][/b] [math]f,-f,\frac{1}{f},-\frac{1}{f}[/math], with a suitable [math]f\in\mathbb{C}[/math].[br]The image [b][i][color=#ff0000]points[/color][/i][/b] lie [b][i][color=#bf9000]point-symmetrically[/color][/i][/b] to the pairs of [b][i][color=#ff0000]points[/color][/i][/b][/size][size=85] [math]\left\{0,\infty\right\},\left\{1,-1\right\},\left\{i,-i\right\}[/math].[br]We call this position the [b][i][color=#0000ff]normal form[/color][/i][/b] of the [b][color=#cc0000]4[/color][/b] [b][i][color=#ff0000]points[/color][/i][/b].[br]If the [b][i][color=#ff0000]points[/color][/i][/b] [math]z_1,z_2,z_3,z_4[/math] [/size][size=85]are [b][i][color=#ff0000]concyclic[/color][/i][/b][/size][size=85], the image [b][i][color=#ff0000]points[/color][/i][/b] lie on one of the [b][i][color=#bf9000]axes[/color][/i][/b] or on the [b][i][color=#bf9000]unit circle[/color][/i][/b].[/size][size=85][br]If the [b][i][color=#ff0000]points[/color][/i][/b] are [b][i][color=#bf9000]mirror-inverted[/color][/i][/b] on [b][color=#cc0000]2[/color][/b] [b][i][color=#0000ff]orthogonal[/color][/i][/b] [b][i][color=#ff0000]circles[/color][/i][/b], the image [b][i][color=#ff0000]points[/color][/i][/b] are also [/size][size=85][b][i][color=#bf9000]mirror-inverted[/color][/i][/b][/size][size=85] on the [b][i][color=#0000ff]angle bisectors[/color][/i][/b]. [br]You can then also place them [/size][size=85][b][i][color=#bf9000]mirror-inverted[/color][/i][/b][/size][size=85] on the axes - this position is also called [b][i][color=#0000ff]normal form[/color][/i][/b].[/size][size=85][br]If the original [b][i][color=#ff0000]points[/color][/i][/b] have a [b][i][color=#0000ff]harmonic position[/color][/i][/b], the image [b][i][color=#ff0000]points[/color][/i][/b] are the intersections of the [b][i][color=#bf9000]unit circle[/color][/i][/b] with the [b][i][color=#0000ff]angle bisectors[/color][/i][/b]. [br][/size][size=85]For the [b][i][u][color=#cc0000]proof[/color][/u][/i][/b] we construct[/size][size=85] [b][color=#cc0000]3[/color][/b] [/size][size=85]pairwise [b][i][color=#0000ff]orthogonal[/color][/i][/b] [b][i][color=#ff0000]circles[/color][/i][/b] in such a way that the original [b][i][color=#ff0000]points[/color][/i][/b][/size][size=85] lie [b][i][color=#bf9000]point-symmetrically[/color][/i][/b] to the[br]intersection [b][i][color=#ff0000]point pairs[/color][/i][/b][/size][size=85]in the sense of [b][i][color=#0000ff]Möbius geometry[/color][/i][/b].[/size][size=85][size=50]Geometrically, this construction is quite complex. Mathematically, one finds the connection with the help of the [br][b]LIE[/b] algebra of the [b][i][color=#0000ff]Möbius group[/color][/i][/b] by a very simple and short calculation (see below!).[br]The actual idea for both the construction and the calculation is based on the principle of [b][i][color=#ff00ff]repeated symmetrisation[/color][/i][/b]![/size][/size][size=85][b][i][u][color=#cc0000][br][br]For the construction:[/color][/u][/i][/b] The [b][color=#cc0000]4[/color][/b] [b][i][color=#ff0000]points[/color][/i][/b] can be divided into [b][color=#cc0000]2[/color][/b] pairs of [b][i][color=#ff0000]points[/color][/i][/b] in [b][color=#cc0000]3[/color][/b] different ways.[br][/size][size=85]For example, [/size][size=85]determine the [b][i][color=#ff0000]circles[/color][/i][/b][/size][size=85] [math]c_{\left\{123\right\}},c_{\left\{124\right\}}[/math] [/size][size=85]and their [b][i][color=#0000ff]bisectors[/color][/i][/b] [/size][size=85][math]cw_{\left\{12\right\}}[/math] and [math]cw'_{\left\{12\right\}}[/math], [/size][size=85]as well as the [b][i][color=#ff0000]circles[/color][/i][/b][/size][size=85][br] [math]c_{\left\{341\right\}},c_{\left\{342\right\}}[/math] [/size][size=85]with the corresponding [b][i][color=#0000ff]angle bisector[/color][/i][/b] [b][i][color=#ff0000]circles[/color][/i][/b][/size][size=85] [math]cw_{\left\{34\right\}}[/math] and [math]cw'_{\left\{34\right\}}[/math] [/size][size=85]for the pairs of [b][i][color=#ff0000]points[/color][/i][/b][/size][size=85] [math]\left\{z_1,z_2\right\},\left\{z_3,z_4\right\}[/math]. [/size][size=85][br][b][color=#cc0000]2[/color][/b] [/size][size=85]of these[/size][size=85] [b][color=#cc0000]4[/color][/b] [/size][size=85][b][i][color=#0000ff]angle bisector[/color][/i][/b] [b][i][color=#ff0000]circles[/color][/i][/b] do not intersect;[/size][size=85][br][size=85]to the other two [b][i][color=#ff0000]circles[/color][/i][/b] you construct their [b][i][color=#0000ff]angle bisector[/color][/i][/b] [b][i][color=#ff0000]circles[/color][/i][/b] (yellow in the applet!).[/size][/size][size=85][br]This construction can be done in three different ways, whereby for [/size][size=85][b][color=#cc0000]2[/color][/b][/size][size=85] different ways,[br] [/size][size=85][b][color=#cc0000]2[/color][/b][/size][size=85] of the resulting [/size][size=85][b][i][color=#0000ff]angle bisector[/color][/i][/b] [b][i][color=#ff0000]circles[/color][/i][/b][/size][size=85] are identical![/size][size=85][br]The result is [/size][size=85][b][color=#cc0000]3[/color][/b][/size][size=85] [b][i][color=#0000ff]orthogonal[/color][/i][/b] [b][i][color=#ff0000]circles.[/color][/i][/b] [/size][size=85]Their intersection [b][i][color=#ff0000]point[/color][/i][/b] pairs are mapped with a [b][i][color=#0000ff]Möbius transformation[/color][/i][/b][br]to the given pairs of [b][i][color=#ff0000]points[/color][/i][/b][/size][size=85] [math]\left\{0,\infty\right\}[/math], [math]\left\{1,-1\right\}[/math] and [math]\left\{i,-i\right\}[/math].[br][br][/size][size=85][b][i][u][color=#cc0000]For the calculation:[/color][/u][/i][/b] [/size][size=85][b][color=#cc0000]2 [/color][/b][/size][size=85][b][i][color=#ff0000]points[/color][/i][/b][/size][size=85] (e.g.[/size][size=85] [math]z_1,z_2[/math]) [size=85]determine an [b][i][color=#ff0000]elliptical pencil of circles[/color][/i][/b] whose [b][i][color=#ff0000]circles[/color][/i][/b] can be seen as [br]trajectories of a [b][i][color=#0000ff]Möbius motion[/color][/i][/b]. [/size][/size][size=85]The corresponding infinitesimal movement is briefly indicated as[/size][size=85] [math]\left(1\wedge2\right)[/math].[br][/size][size=85]Accordingly, let[/size][size=85] [math]\left(3\wedge4\right)[/math] [/size][size=85]be interpreted.[/size][size=85] The [b]LIE[/b]-product [math]\left[\left(1\wedge2\right),\left(3\wedge4\right)\right][/math] [/size][size=85]belongs to an infinitesimal movement,[/size][size=85]i.e. [b][i][color=#ff0000]elliptical circular motion[/color][/i][/b], [/size][size=85]whose pencil [b][i][color=#ff0000]points[/color][/i][/b] lie [b][i][color=#0000ff]harmonically[/color][/i][/b] with those of [/size][size=85] [math]\left(1\wedge2\right)[/math] and [math]\left(3\wedge4\right)[/math].[br]The [b]LIE[/b]-products [math]\left[\left(1\wedge2\right),\left(3\wedge4\right)\right],\left[\left(1\wedge3\right),\left(2\wedge4\right)\right],\left[\left(1\wedge4\right),\left(2\wedge3\right)\right][/math] are pairwise "[b][i][color=#0000ff]orthogonal[/color][/i][/b]", [size=85]which geometrically[br]means that the base [b][i][color=#ff0000]point[/color][/i][/b] pairs lie on[/size] [b][color=#cc0000]3[/color][/b] [/size][size=85]pairwise [b][i][color=#0000ff]orthogonal[/color][/i][/b] [/size][size=85][b][i][color=#ff0000]circles[/color][/i][/b][/size][size=85]and the [/size][size=85][b][color=#cc0000]4[/color][/b][/size][size=85] given [b][i][color=#ff0000]points[/color][/i][/b] lie [b][i][color=#0000ff]harmonically[/color][/i][/b], [br]i.e. [b][i][color=#bf9000]point-symmetrically[/color][/i][/b], to these [b][i][color=#ff0000]base point[/color][/i][/b] pairs![/size]