[table][tr][td][url=https://www.geogebra.org/m/nzfg796n#material/ys3tbadp][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]07.08.2023[/b][/i][/color])[/size][/td][/tr][/table]

[b][color=#ff7700][size=50][right]07.08.2023[/right][/size][/color][/b][size=85]Four different [b][i][color=#ff7700]points[/color][/i][/b] in the complex plane can always be mapped by a suitable [b][i][color=#0000ff]Möbius transformation[/color][/i][/b][br]to [b][color=#a61c00]4[/color][/b] [b][i][color=#ff7700]points[/color][/i][/b] in [b][i]normal form[/i][/b] [math]f,-f,\frac{1}{f},-\frac{1}{f}[/math] with [math]f\notin\left\{0,\infty,1,-1,i,-i\right\}[/math].[br]Depending on the order of the [b][i][color=#ff7700]points[/color][/i][/b] the [b][i][color=#6aa84f]complex[/color][/i][/b] [b][i][color=#0000ff]double ratio[/color][/i][/b] [math]dv=\frac{f-\frac{1}{f}}{-f-\frac{1}{f}}\cdot\frac{-f+\frac{1}{f}}{f+\frac{1}{f}}=\frac{\left(f^2-1\right)^2}{\left(f^2+1\right)^2}[/math].[br][b][i]Absolutely invariant [/i][/b]is [math]J_{\left\{abs\right\}}=\frac{1}{27}\cdot\left(\frac{dv+1}{dv-1}\right)^2\cdot\left(\frac{dv-2}{dv}\right)^2\cdot\left(2\cdot dv-1\right)^2[/math].[br][/size][list][*][size=85][b][color=#a61c00]4 [/color][/b]distinct [b][i][color=#ff0000]points[/color][/i][/b] are the [b][i][color=#00ff00]foci[/color][/i][/b] of [b][i][color=#6aa84f]confocal[/color][/i][/b] [b][i][color=#ff7700]bicircular quartics[/color][/i][/b],[/size][size=85][br]if their [b][i]absolute invariant[/i][/b] [math]J_{\left\{abs\right\}}[/math] is real.[/size][/*][*][size=85]If [math]J_{\left\{abs\right\}}[/math] is real and non-negative, the [b][i][color=#00ff00]foci[/color][/i][/b] are [b][i][color=#ff0000]concyclic[/color][/i][/b][b][i][color=#ff0000][/color][/i][/b].[/size][/*][*][size=85]If [math]J_{\left\{abs\right\}}[/math] is real and not positiv, then the [b][i][color=#00ff00]foci[/color][/i][/b] in [b][color=#a61c00]2[/color][/b] pairs lie [b][i][color=#e69138]mirror-inverted[/color][/i][/b] [br]on [b][i][color=#a61c00]2 [/color][color=#0000ff]orthogonal[/color][/i][/b] [b][i][color=#ff0000]circles[/color][/i][/b].[/size][/*][*][size=85]If [math]J_{\left\{abs\right\}}=0[/math], then both are true: the [b][i][color=#00ff00]foci[/color][/i][/b] lie [b][i][color=#0000ff]harmonically[/color][/i][/b].[br][/size][/*][/list][size=85]In the appplet above, the [b][i]absolute invariant[/i][/b] is real if [math]f[/math] lies on one of the axes, or on the [b][i][color=#e69138]unit circle[/color][/i][/b],[br]or on one of the [b][i][color=#0000ff]bisecting[/color][/i][/b] [b][i][color=#ff0000]circles[/color][/i][/b] or [b][i][color=#ff0000]straight lines[/color][/i][/b]![br]The highest [b][i][color=#e69138]symmetry[/color][/i][/b] is in the [b][i][color=#9900ff]tetrahedron case[/color][/i][/b]: [math]f[/math] coincides with one of the [b][i][color=#ff0000]points[/color][/i][/b], [br]i[/size][size=85]n which [b][i][color=#a61c00]3[/color][/i][/b] [b][i][color=#0000ff]angle bisectors[/color][/i][/b] intersect: [/size][size=85] [math]J_{\left\{abs\right\}}=-1[/math].[/size]