[table][tr][td][url=https://www.geogebra.org/m/nzfg796n#material/dnq7xvge][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] [/url][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]

[size=85][b][color=#cc0000]4[/color][/b] [/size][size=85]different complex [b][i][color=#ff0000]points[/color][/i][/b] [/size][size=85] [math]z_1,z_2,z_3,z_4\in\mathbb{C}\cup\left\{\infty\right\}[/math] [/size][size=85]are characterised [b][i][color=#0000ff]möbius-geometrically[/color][/i][/b] [br]by their [b][i][color=#0000ff]complex[/color][/i][/b] [b][i][color=#9900ff]cross product[/color][/i][/b][/size][list][*][size=85] [math]d=Dv\left(z_1,z_2,z_3,z_4\right)=\frac{z_3-z_1}{z_3-z_2}\cdot\frac{z_4-z_2}{z_4-z_1}[/math][br][/size][/*][/list][size=85][b][color=#cc0000]2[/color][/b] [/size][size=85]quadruples of [b][i][color=#ff0000]points[/color][/i][/b] with identical[/size][size=85][b][i][color=#9900ff] cross product[/color][/i][/b][/size][size=85]can be mapped by a [b][i][color=#0000ff]Möbius transformation[/color][/i][/b][/size][size=85] [math]z\mapsto\frac{a\cdot z+b}{c\cdot z+d}[/math][br]: [/size][size=85]one onto the other, but depending on the order.[/size][size=85][br]Independent of the order is the [b][i]absolute invariant[/i][/b] of [b][i][color=#cc0000]four[/color][/i][/b] [b][i][color=#ff0000]points[/color][/i][/b]:[/size][size=85][br][list][*][math]J_{\left\{abs\right\}}=\frac{1}{27}\cdot\left(\frac{d+1}{d-1}\right)^2\cdot\left(\frac{d-2}{d}\right)^2\cdot\left(2\cdot d-1\right)^2[/math][br][/*][/list][/size][size=85]For an [b][i][color=#38761d]elliptic function[/color][/i][/b] with the [b][i][color=#9900ff]differential equation[/color][/i][/b][/size][size=85] [math]\left(g'\right)^2=c\cdot\left(g-f_1\right)\cdot\left(g-f_2\right)\cdot\left(g-f_3\right)\cdot\left(g-f_4\right)[/math] [br][/size][size=85]and different[/size][size=85] [b][i][color=#00ff00]focal points[/color][/i][/b] [math]f_1,f_2,f_3,f_4[/math] [size=85]its [b][i]absolute invariant[/i][/b] [/size][br][/size][size=85]is also called the [b][i]absolute invariant[/i][/b] of the [b][i][color=#38761d]elliptic function[/color][/i][/b].[/size]

[size=85]In the applet above on the right,[/size][size=85] a [b][i][color=#0000ff]Möbiustransformation[/color][/i][/b] [size=85]has been mapped[/size] [math]z_1[/math] to 0, [math]z_2[/math] to [math]\infty[/math] and [math]\left\{z_3,z_4\right\}[/math] to [math]\left\{z'_3,z'_4\right\}[/math]. [b][i][u][color=#cc0000][br]special positions:[/color][/u][br][/i][u]I.:[/u][/b] [/size][size=85]The [b][i][color=#ff0000]circles[/color][/i][/b][/size][size=85] [math]c_{\left\{123\right\}}[/math] through [math]z_1,z_2,z_3[/math] and [math]c_{\left\{124\right\}}[/math] through [math]z_1,z_2,z_4[/math] [/size][size=85]are [b][i][color=#0000ff]orthogonal[/color][/i][/b] exactly,[/size][size=85],[br] [/size][size=85]if the [b][i][color=#9900ff]double ratio[/color][/i][/b] [/size][size=85] [math]Dv\left(z_1,z_2,z_3,z_4\right)=\frac{z_3-z_1}{z_3-z_2}\cdot\frac{z_4-z_2}{z_4-z_1}[/math] [/size][size=85]is purely imaginary[/size][size=85].[br] [/size][size=85]In the picture on the right, this is the case exactly when the [b][i][color=#ff0000]straight lines[/color][/i][/b][/size][size=85] [math]z_3,z_2,\infty[/math] and [math]z_4,z_2,\infty[/math] [/size][size=85]are [b][i][color=#0000ff]orthogonal[/color][/i][/b],[/size][size=85][br] i.e. exactly then, when [/size][size=85] [math]z_2[/math] [/size][size=85]on the [b]THALES[/b] [b][i][color=#ff0000]circle[/color][/i][/b][/size][size=85] lies over[/size][size=85] [math]z_3,z_4[/math]![br] The [b][i][color=#ff0000]circles[/color][/i][/b] [math]c_{\left\{134\right\}}[/math] through [math]z_1,z_3,z_4[/math] and [math]c_{\left\{234\right\}}[/math] through [math]z_2,z_3,z_4[/math] [/size][size=85]are then also [b][i][color=#0000ff]orthogonal[/color][/i][/b].[/size][size=85][br] The [b][i]absolute invariant[/i][/b][/size][size=85] [math]J_{abs}[/math] [/size][size=85]does not reveal any special feature.[/size][size=85][br][b][br]II.:[/b] [/size][size=85]The pairs of [b][i][color=#ff0000]points[/color][/i][/b][/size][size=85] [math]\left\{z_1,z_2\right\}[/math] and [math]\left\{z_3,z_4\right\}[/math] [/size][size=85]lie [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] exactly then,[/size][size=85][br] when [math]j_1:=\frac{d+1}{d-1}\in i\mathbb{R}[/math]; [/size][size=85]this is the case exactly if[/size][size=85] [math]\left|d\right|=1[/math] holds;[br] [/size][size=85]and this is the case exactly if the [b][i]absolute invariant[/i][/b][/size][size=85] [math]J_{abs}[/math] is real and not positiv: [math]J_{abs}\in\mathbb{R}[/math] and [math]J_{abs}\le0[/math].[br][i][u][color=#cc0000][br]reason[/color][/u][/i]: [math]\left|d\right|=1\Longleftrightarrow\frac{d+1}{d-1}\in i\mathbb{R}[/math] [/size][size=85] is the theorem of [b]THALES[/b], see below.[/size][size=85]A [b][i][color=#ff0000]circle[/color][/i][/b] to which the [b][i][color=#ff0000]points[/color][/i][/b][/size][size=85] [math]z_1,z_2[/math] [/size][size=85]are [b][i][color=#bf9000][br] mirror-images[/color][/i][/b][/size][size=85], [/size][size=85]must be a [b][i][color=#ff0000]circle[/color][/i][/b] from the [b][i][color=#ff0000]hyperbolic[/color][/i][/b] [b][i][color=#ff0000]pencil of circles[/color][/i][/b] [br] around[/size][size=85] [math]z_1[/math] and [math]z_2[/math] ([/size][size=85]the [b][i][color=#bf9000]inversion[/color][/i][/b] at this [b][i][color=#ff0000]circle[/color][/i][/b] interchanges the two [b][i][color=#ff0000]points[/color][/i][/b]![/size][size=85])[br] [size=85]Exactly one [b][i][color=#ff0000]circle[/color][/i][/b] each from this [b][i][color=#ff0000]pencil of circles[/color][/i][/b] goes t[/size]hrough [math]z_3[/math] and through [math]z_4[/math][/size][size=85]. [br] [math]z_1[/math] and [math]z_2[/math] [/size][size=85]can therefore only lie [b][i][color=#bf9000]mirror-inverted[/color][/i][/b] to a [b][i][color=#ff0000]circle[/color][/i][/b] through[/size][size=85] [math]z_3,z_4[/math], [/size][size=85]if the two [b][i][color=#ff0000]circles[/color][/i][/b] of the [br] [b][i][color=#ff0000]pencil[/color][/i][/b] are identical[/size][size=85]. [/size][size=85]In the applet one obtains on the right[/size][size=85] [math]d=Dv\left(z_1,z_2,z_3,z_4\right)=Dv\left(0,\infty,z'_3,z'_4\right)=\frac{z'_3}{z'_4}[/math].[br] [math]\left|d\right|=1[/math] [/size][size=85] is therefore exactly the case if [/size][size=85] [math]z'_3,z'_4[/math] [/size][size=85]lie around 0 on the same [b][i][color=#ff0000]concentric circle[/color][/i][/b].[/size][size=85] [br] The [b][i][color=#bf9000]inversion[/color][/i][/b] at this [/size][size=85][size=85][b][i][color=#ff0000]circle[/color][/i][/b][/size][/size][size=85] interchanges 0 and [math]\infty[/math], the [b][i][color=#0000ff]angle bisectors[/color][/i][/b] of the [b][i][color=#ff0000]straight lines[/color][/i][/b] [math]0,z'_3[/math] and [math]0,z'_4[/math][br] are [b][i][color=#0000ff]orthogonal[/color][/i][/b] to the concentric [/size][size=85][size=85][b][i][color=#ff0000]circle[/color][/i][/b][/size][/size][size=85], mirrored on them [math]z'_3[/math] and [math]z'_4[/math] are interchanged.[br] We explain the connection with the [b][i]absolute invariant[/i][/b] [math]J_{abs}[/math] below![/size]

[size=85][b]III.: [color=#cc0000] 4[/color][/b] [b][i][color=#ff0000]points [/color][/i][/b][size=85]lie on a [b][i][color=#ff0000]circle[/color][/i][/b] exactly when their [b][i][color=#9900ff]double ratio[/color][/i][/b] is real; [/size][br] [/size][size=85]this is the case exactly if the [b][i]absolute invariant[/i][/b] of the [b][color=#cc0000]4[/color][/b] [b][i][color=#ff0000]points[/color][/i][/b] is real and non-negative![/size][size=85][br][u][i][color=#e06666]reason: [/color][/i][/u] [/size][size=85]According to the [b][i][color=#38761d]Peripheral Angle Theorem[/color][/i][/b], the [b][i][color=#9900ff]double ratio[/color][/i][/b] is real exactly if,[/size][size=85][br] if the [b][i][color=#ff0000]points[/color][/i][/b] lie on a circle. [size=85]This property is independent of the order of the [b][i][color=#ff0000]points[/color][/i][/b].[/size][br] For real [math]d[/math] is [math]J_{abs}[/math] real and non-negative![/size]

[size=85][b][i][u][color=#cc0000]Under which condition is the absolute invariant[/color][/u][/i][/b][/size] [math]J_{abs}[/math] [size=85][b][i][u][color=#cc0000]real?[/color][/u][/i][/b][br]The set of [b][color=#cc0000]6[/color][/b] relative [b][i]invariants[/i][/b] [math]j_{1/2}=\pm\frac{d+1}{d-1}[/math], [math]j_{3/4}=\pm\frac{d-2}{d}[/math] and [math]j_{5/6}=\pm\left(2\cdot d-1\right)[/math] is invariant under[br][/size][size=85]the finite group of [b][i][color=#bf9000]involutory[/color][/i][/b] [b][i][color=#0000ff]Möbius transformations[/color][/i][/b][/size][size=85] [math]\left\{\pm id,\pm A,\pm B\right\}[/math] with [math]Az=\frac{3+z}{z-1}[/math], [math]Bz=\frac{3-z}{z+1}[/math].[br][/size][size=85]For each of the [b][color=#cc0000]6[/color][/b] relative [b][i]invariants[/i][/b] holds [/size][list][*][size=85][math]27\cdot J_{abs}=j^2\cdot\left(\frac{3+j}{j-1}\right)^2\cdot\left(\frac{3-j}{j+1}\right)^2=j^2\cdot\left(\frac{9-j^2}{j^2-1}\right)[/math][br][/size][/*][/list][size=85]This [b][i]cubic equation[/i][/b] in [math]j^2[/math] has only real coefficients.[br][math]J_{abs}\ge0[/math] [/size][size=85]is true if and only if[/size][size=85] [math]d[/math] is real, [size=85]i.e. if the [b][i][color=#ff0000]points[/color][/i][/b] are [b][i][color=#ff0000]concyclic[/color][/i][/b][/size].[br][math]J_{abs}\le0[/math] [/size][size=85]is true if and only if[/size][size=85] [math]j\in i\mathbb{R}[/math] [/size][size=85]for any of the six relative invariants. [br]By changing [/size][size=85]the order of the [b][i][color=#ff0000]points[/color][/i][/b] can be achieved[/size][size=85] [math]\frac{d+1}{d-1}\in i\mathbb{R}[/math] [/size][size=85]i.e. mirror-image position on [b][color=#cc0000]2[/color][/b] [b][i][color=#0000ff]orthogonal[/color][/i][/b] [b][i][color=#ff0000]circles[/color][/i][/b].[/size]

[size=85][b][u][color=#cc0000]2[/color][/u][i][u][color=#cc0000] very special positions:[/color][/u][/i][/b][br][br][/size][size=85]If the [b][color=#cc0000]4[/color][/b] [b][color=#ff0000]points[/color][/b] are different and is[/size][size=85] [math]J_{abs}=0[/math], [/size][size=85]then both are valid: the [b][i][color=#ff0000]points[/color][/i][/b] are [b][i][color=#ff0000]concyclic [/color][/i][/b][/size][size=85]and lie [b][i][color=#bf9000]mirror-image[/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].[/size][size=85][br] [b][i][u][color=#cc0000]Example:[/color][/u][/i][/b] [/size][size=85]the intersections of the 1st and 2nd [b][i][color=#0000ff]bisectors[/color][/i][/b] with the [b][i][color=#bf9000]unit circle[/color][/i][/b].[/size][size=85][br][br] [math]J_{abs}=-1[/math] [/size][size=85]is present for [b][color=#cc0000]4[/color][/b] [b][i][color=#ff0000]points[/color][/i][/b] whose [b][i][color=#9900ff]double ratio[/color][/i][/b] [/size][size=85] [math]\Large{d=e^{\frac{\pi}{3}\cdot i}}[/math] [/size][size=85]is independent of the order[/size][size=85].[br][/size][size=85]If one projects the [b][i][color=#ff0000]points[/color][/i][/b] [b][i][color=#0000ff]stereographically[/color][/i][/b] onto the [b][i][color=#980000]unit sphere[/color][/i][/b], one obtains a [b][i][color=#0000ff]regular[/color][/i][/b] [b][i][color=#38761d]tetrahedron[/color][/i][/b] on the [b][i][color=#980000]sphere[/color][/i][/b].[/size]