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

[size=85][b][i][color=#9900ff]differential equation[/color][/i][/b] of a [b][i][color=#ff0000]parabolic pencil of circles[/color][/i][/b]:[br][/size][list][*][size=85][math]p'=\frac{p-f}{u}\cdot\left(p-f\right)=\frac{1}{u}\cdot\left(p-f\right)^2[/math][br][/size][/*][/list][size=85]The complex vector [math]u[/math] indicates the direction in which the [b][i][color=#ff0000]circles[/color][/i][/b] in [b][i][color=#00ff00]f[/color][/i][/b] touch each other.[br][b][i][u][color=#cc0000]solution:[/color][/u][/i][/b][br][/size][list][*][size=85][math]p(z)=f-\frac{u}{z}[/math] , es ist [math]p'(z)=\frac{1}{u}\cdot\left(p\left(z\right)-f\right)^2[/math] . [br][/size][/*][/list][size=85][b][i][color=#9900ff]differential equation[/color][/i][/b] of a [b][i][color=#ff0000]elliptic pencil of circles[/color][/i][/b]:[br][/size][list][*][size=85][math]p'=\frac{\left(p-f_1\right)\cdot\left(p-f_2\right)}{f_2-f_1}[/math][br][/size][/*][/list][size=85][b][i][u][color=#cc0000]solution:[/color][/u][/i][/b][br][/size][list][*][math]z\mapsto e^z[/math] [size=85][/size][size=85]maps the [b][i][color=#ff0000]parallels[/color][/i][/b] to the [math]x[/math]-axis to the [b][i][color=#ff0000]rays[/color][/i][/b] through [b][i][color=#00ff00]f[/color][/i][/b], [/size][/*][*][size=85]the parallels to the [math]y[/math]-axis on the concentric circles around [math]0[/math]. [/size][br][size=85]the [b][i][color=#0000ff]m[/color][/i][/b][/size][size=85][b][i][color=#0000ff]oebiustransformation[/color][/i][/b][/size] [math]Tz=\frac{f_2\cdot\left(z-f_1\right)}{z-f_2}[/math] [size=85]maps [/size][math]0[/math] [size=85]to[/size] [math]f_1[/math] [size=85]and[/size] [math]\infty[/math] [size=85]to[/size] [math]f_2[/math][size=85]. [br]for [/size][math]p\left(z\right)=\frac{f_2\cdot\left(e^z-f_1\right)}{e^z-f_2}[/math][size=85] [/size][size=85] this gives[/size][size=85] [/size][math]p'\left(z\right)=\frac{1}{f_2-f_1}\cdot\left(p\left(z\right)-f_1\right)\cdot\left(p\left(z\right)-f_2\right)[/math].[size=85][br][/size][/*][/list][size=85][b][i][u][color=#cc0000]The geometric idea[/color][/u][/i][/b] uses the [math]\hookrightarrow[/math] [url=https://en.wikipedia.org/wiki/Inscribed_angle_theorem][b][i][color=#0000ff]inscribed angle theorem[/color][/i][/b][/url] and the properties of the [b][i][color=#ff0000]tangents[/color][/i][/b] in the intersections [br]of an segment of a [b][i][color=#ff0000]circle [/color][/i][/b]with the [b][i][color=#ff0000]circle[/color][/i][/b].[br][b][i][u][color=#cc0000]And:[/color][/u][/i][/b] The multiplication of two complex numbers [math]u=r\cdot e^{i\cdot\varphi}[/math] and [math]v=\rho\cdot e^{i\cdot\psi}[/math] can be [br]interpreted as a [b][i][color=#38761d]stretchrotation[/color][/i][/b]: [math]u\cdot v=r\cdot\rho\cdot e^{i\cdot\left(\varphi+\psi\right)}[/math].[/size]