[right][i][b][size=50][color=#ff7700]12. April 2020 [/color][/size][/b][/i][b][size=50]Diese Aktivität ist eine Seite des[/size][/b][i][b][size=50] [color=#980000][url=https://www.geogebra.org/m/kCxvMbHb]geogebra-books Moebiusebene[br][/url][/color][/size][/b][/i][size=85][size=50]siehe auch [u][color=#a64d79][i][b][color=#0000ff][/color][url=https://www.geogebra.org/m/uufqfu6b][color=#0000ff]bizirkularer Ostergruss[/color]![/url][/b][/i][/color][/u][/size][/size][br][size=85][/size][/right][size=85][br]Diese 2-teilige [color=#ff00ff][i][b]Darboux[/b][/i][/color] [i][b][color=#0000ff]Cyclide [/color][/b][/i]ist [color=#BF9000][i][b]symmetrisch[/b][/i][/color] zu den 3 [color=#3c78d8][i][b]Koordinatenebenen[/b][/i][/color] und zur [color=#999999][i][b]Einheitskugel[/b][/i][/color]![br]Überdies ist sie [color=#BF9000][i][b]rotationssymmetrisch[/b][/i][/color] zur [math]x[/math]-Achse.[br]Jede [color=#ff0000][i][b]Kugel[/b][/i][/color] - und dazu gehören [color=#0000ff][i][b]möbiusgeometrisch[/b][/i][/color] auch die [color=#3c78d8][i][b]Ebenen[/b][/i][/color] - schneidet diese [color=#ff7700][i][b]bizirkulare Fläche 4.-ter Ordnung[/b][/i][/color] in einer [color=#ff7700][i][b]bizirkularen Quartik[/b][/i][/color]![br]Die [color=#BF9000][i][b]Rotationssymmetrie[/b][/i][/color] erlaubt uns die Darstellung der Fläche in [color=#999999][i][b]Parameter-Form[/b][/i][/color].[br]Leider gelingt uns die Parameterdarstellung für den Fall [math]C_x\ne B_y[/math] noch(?) nicht![br][br]Die [color=#f6b26b][i][b]Höhenlinien [/b][/i][color=#000000]in [b]3D[/b][/color][/color] [math]\left(x^2+y^2+z^2\right)^2-2\cdot A_x\cdot x^2-2\cdot B_y\cdot y^2-2\cdot C_z\cdot z^2+1=0[/math] mit [math]C_z=B_y[/math] und [math]z=h_z[/math] werden als [br][color=#f1c232][i][b]implizite Kurven[/b][/i][/color] mit [color=#ff7700][i][b]Spur[/b][/i][/color] gezeichnet und verschwinden nach Änderungen der graphischen Einstellungen - wie zB. durch [i][b]Zoomen[/b][/i]![/size]

[size=85]In der [math]xy[/math]-Ebene [math]\mathbb{C}[/math][color=#0000ff][i][b] implizit[/b][/i][/color]: [math]\left(x^2+y^2\right)^2-2\cdot A_x\cdot x^2-2\cdot B_y\cdot y^2+1=0[/math] mit [math]A_x\ge1[/math] und [math]B_y\le-1[/math].[br][color=#0000ff][i][b]Parameterdarstellung[/b][/i][/color]:[br][list][*][math]\mathbf{bizQu}_{\left\{\pm\right\}}\left(u\right):=T^{-1}\left(\frac{1}{N_{\left\{\pm\right\}}\left(u\right)}\cdot \mathbf{ellipse}\left(u\right)\right)=\frac{\frac{1}{N_{\left\{\pm\right\}}\left(u\right)}\cdot \mathbf{ellipse}\left(u\right)+1}{1-\frac{1}{N_{\left\{\pm\right\}}\left(u\right)}\cdot \mathbf{ellipse}\left(u\right)}[/math][br] mit [math]N_{\left\{\pm\right\}}\left(u\right):=1\pm\sqrt{1-a^2\cdot cos^2\left(u\right)-b^2\cdot sin^2\left(u\right)}[/math][br] [br] und [math]\mathbf{ellipse}(u):=a\cdot cos\left(u\right)+i\cdot b\cdot sin\left(u\right)[/math], [math]-\pi\le u\le\pi[/math].[/*][br][*]Dabei ist [math]a=\frac{2\cdot T(s_x)}{T(s_x)^2+1}[/math] mit dem [color=#ff7700][i][b]Scheitelwert[/b][/i][/color] auf der [math]x[/math]-Achse [math]s_x=\sqrt{A_x+\sqrt{A_x^2-1}}[/math][/*][br][*] und [math]b=\frac{2\cdot TSE}{TSE^2+1}[/math] mit [math]TSE=\mbox{Imaginärteil}\left(T\left(SE\right)\right) [/math], [br][math]SE=\sqrt{\frac{B_y-1}{B_y-A_x}}+i\cdot\sqrt{\frac{1-A_x}{B_y-A_x}}[/math] ist ein [color=#ff7700][i][b]Schnittpunkt[/b][/i][/color] mit dem [i][b]Einheitskreis[/b][/i] [math]x^2+y^2=1[/math].[/*][br][*] Die [color=#0000ff][i][b]Möbiustransformation[/b][/i][/color] [math]T\left(z\right)=\frac{z-1}{z+1}[/math] mit [math]-1\mapsto\infty\mapsto1\mapsto0\mapsto-1[/math] bildet den [i][b]Einheitskreis[/b][/i] auf die [math]y[/math]-Achse ab [br]und die [math]x[/math]-Achse auf sich. Die Inverse ist [math]T^{-1}\left(z\right):=\frac{z+1}{1-z}[/math][/*][/list][center] 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[/img][/center][/size]