[br][table][tr][td][url=https://www.geogebra.org/m/nzfg796n#material/qmknpdwx][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/nzfg796n#material/hhnfvjkv][color=#0000ff][u][i][b]elliptic functions & bicircular quartics & . . .[/b][/i][/u][/color][/url]([color=#ff7700][i][b]30.04.2023[/b][/i][/color])[/size][/td][/tr][/table][size=50][i][b][size=50][right][size=50]this activity is also a page of[/size] [color=#980000]geogebra-book[/color] [url=https://www.geogebra.org/m/xtueknna][color=#0000ff][u]geometry of some complex functions[/u][/color][/url] [color=#ff7700]october 2021[/color][/right][/size][/b][/i][/size]

[size=85]An [b][i][color=#ff0000]elliptical pencil of circles[/color][/i][/b] consists of all [b][i][color=#ff0000]circles[/color][/i][/b] through [b][color=#cc0000]2[/color][/b] base points, which we also call [b][i][color=#00ff00]focal points[/color][/i][/b].[br]The [b][i][color=#0000ff]orthogonal[/color][/i][/b] [color=#ff0000]circles[/color] form the polar [b][i][color=#ff0000]hyperbolic pencil of circles[/color][/i][/b].[br]In the applet above, the pencil of rays through [b][color=#0000ff]w[sub]0[/sub][/color][/b] is an [b][i][color=#ff0000]elliptical pencil of circles[/color][/i][/b], the 2nd [b][i][color=#00ff00]focal point[/color][/i][/b] is [math]\infty[/math]. [br]The [b][i][color=#ff0000]concentric circles[/color][/i][/b] around [b][color=#0000ff]w[sub]0[/sub][/color][/b] are the polar [b][i][color=#ff0000]hyperbolic pencil[/color][/i][/b].[br]The [b][i][color=#ff0000]circles[/color][/i][/b] arise from the [b][i][color=#ff00ff]axis-parallel straight lines[/color][/i][/b] as images under the complex function[/size][size=85][size=85][br][list][*][math]g(z)=w_0+e^{x+i\cdot y}=e^x\cdot\left(\cos\left(y\right)+i\cdot \sin\left(y\right)\right)[/math] für [math]x=\mathbf{const}[/math] oder [math]y=\mathbf{const}[/math][/*][/list][/size][/size][size=85][size=85][br]In the applet below, [/size][/size][size=85][size=85][color=#0000ff][b]w[sub]0[/sub][/b][/color][/size][/size][size=85][size=85] and [b][color=#0000ff]w[/color][/b][math]_{\infty}[/math] are the [b][i][color=#00ff00]focal points[/color][/i][/b].[br]The [b][i][color=#ff0000]circles[/color][/i][/b] are images of the [b][i][color=#ff00ff]axis-parallel straight lines[/color][/i][/b] under the complex function[br][/size][/size][list][*][size=85][math]g\left(z\right)=\frac{w_{\infty}\cdot e^z-s_1\cdot w_0}{e^z-s_1},\mbox{ mit }s_1=\frac{w_1-w_{\infty}}{w_1-w_0}[/math][br][/size][/*][/list]

[size=85]In general, [b][i][color=#ff0000]pencils of circles[/color][/i][/b] and their [b][i][color=#9900ff]loxodromes[/color][/i][/b] [br] - i.e. the curves, which intersect the [b][i][color=#ff0000]circles[/color][/i][/b] of the [b][i][color=#ff0000]pencil[/color][/i][/b] at a constant angle - [br]are characterised by a [b][i][color=#38761d]differential equation[/color][/i][/b] and thus by a [b][i][color=#38761d]vector field[/color][/i][/b] of the type[br][list][*][math]g'=c\cdot\left(g-f_1\right)\cdot\left(g-f_2\right)\mbox{ mit }f_1,f_2,c\in\mathbb{C}[/math].[/*][/list]Here the [b][i][color=#38761d]complex solution function[/color][/i][/b] is analytical, or meromorphic. [br]The zeros [math]f_1,f_2[/math], which we call [b][i][color=#00ff00]focal points[/color][/i][/b], can coincide ( - then there is a [b][i][color=#ff0000]parabolic circle pencil[/color][/i][/b] - ).[br]One can interpret the [b][i][color=#ff7700]circles[/color][/i][/b] of a [b][i][color=#ff0000]hyperbolic pencil of circles[/color][/i][/b] dynamically as [b][i]circular waves[/i][/b], which propagate [br]from a [b][i][color=#00ff00]source[/color][/i][/b] in the direction of the [b][i][color=#ff0000]circles[/color][/i][/b] of the [b][i][color=#0000ff]orthogonal[/color][/i][/b] [b][i][color=#ff0000]elliptical pencil[/color][/i][/b] to an [b][i][color=#00ff00]sink[/color][/i][/b]. [br]The [b][i][color=#00ff00]source[/color][/i][/b] and the [b][i][color=#00ff00]sink[/color][/i][/b] are the [b][i][color=#00ff00]focal points[/color][/i][/b] of the [b][i][color=#38761d]wave motion[/color][/i][/b].[br]We call these [b][i][color=#38761d]vector fields[/color][/i][/b] [b][i]linear[/i][/b]. [br]For explanation we refer to the representation of the [b][i][color=#0000ff]Möbius group[/color][/i][/b] by the complex special orthogonal group [b]SO(3,[math]\mathbb{C}[/math])[/b][br]and its [b]LIE [/b]algebra [math]\mathbf{\mathcal{so}\left(3,\mathbb{C}\right)[/math]. [math]\hookrightarrow[/math] [color=#980000][i][b]geogebra-book[/b][/i][/color] [color=#0000ff][i][b]Möbiusebene[/b][/i][/color], special the chap. [url=https://www.geogebra.org/m/kCxvMbHb#chapter/168949][color=#0000ff][u][i][b]Kreisbüschel und lineare Vektorfelder[/b][/i][/u][/color][/url][br][br]If [b][color=#cc0000]2[/color][/b] such [b][i][color=#38761d]vector fields[/color][/i][/b] are superimposed, "[b][i][color=#38761d]quadratic vector fields[/color][/i][/b]" are created whose solution curves [br]can be [b][i][color=#38761d]confocal[/color][/i][/b] [b][i][color=#ff7700]conic sections[/color][/i][/b] or [b][i][color=#38761d]confocal[/color][/i][/b] [b][i][color=#ff7700]bicircular quartics[/color][/i][/b]. [br][b][i][color=#00ff00]Focal points[/color][/i][/b] are the zeros of the [b][i]linear[/i][/b] [b][i][color=#38761d]vector fields[/color][/i][/b].[br]The solution curves in these cases are [b][i][color=#0000ff]angle bisectors[/color][/i][/b] of the intersecting [b][i][color=#ff0000]circles[/color][/i][/b] from the two [b][i][color=#ff0000]pencils of circles[/color][/i][/b].[/size][size=85][br][br]links: [br][math]\hookrightarrow[/math] [color=#980000][i][b]geogebra-book[/b][/i][/color] [color=#0000ff][u][i][b][url=https://www.geogebra.org/m/kCxvMbHb]möbiusebene[/url][/b][/i][/u][/color][br][math]\hookrightarrow[/math] [/size][size=85][size=85][color=#980000][i][b]geogebra-book[/b][/i][/color][/size] [color=#0000ff][u][i][b][url=https://www.geogebra.org/m/fzq79drp]Leitlinien und Brennpunkte[/url][/b][/i][/u][/color][br][br][/size]