[table][tr][td][url=https://www.geogebra.org/m/nzfg796n#material/vz95c3xa][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][/b][/i][/color][color=#ff7700][i][b]06.02.2023[/b][/i][/color])[/size][/td][/tr][/table]

[size=85][color=#9900ff][i][b]Elliptic functions[/b][/i][color=#000000] are meromorphic doubly-periodic complex functions,[br]which of a complex [color=#9900ff][i][b]differential equation[/b][/i][/color] of the type[/color][/color][br][/size][list][*][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] with [math]c\in\mathbb{C}[/math][br][/size][/*][/list][size=85]suffice.[br]The [color=#00ff00][i][b]foci[/b][/i][/color] [math]f_1,f_2,f_3,f_4\in\mathbb{C}\cup\left\{\infty\right\}[/math] are assumed to be different for the actual [color=#9900ff][i][b]elliptic functions[/b][/i][/color].[br]Examples are the [b]Weierstrass[/b] [math]\wp[/math]-functions with the [/size][size=85][color=#9900ff][color=#000000][color=#9900ff][i][b]differential equation[/b][/i][/color][/color][/color][/size][size=85] [math]\left(\wp'\right)^2=\left(\wp-f_1\right)\cdot\left(\wp-f_2\right)\cdot\left(\wp-f_3\right)[/math] [br]- a [color=#00ff00][i][b]focal point[/b][/i][/color] of these functions is [math]\infty[/math], the image on the left with [color=#ff0000][i][b]concyclic[/b][/i][/color] [color=#00ff00][i][b]foci[/b][/i][color=#000000] - [/color][/color][br][color=#9900ff][color=#000000]and the[/color][color=#000000][b]Jacobian[/b][/color][i][b] elliptic functions[/b][/i][/color] [b]sn[/b], [b]dn[/b], [b]cn[/b], ... .[br] [math]\hookrightarrow[/math] [color=#0000ff][u][b][url=https://de.wikipedia.org/wiki/Elliptische_Funktion]elliptic functions[/url][/b][/u][/color] ; [math]\hookrightarrow[/math] [color=#0000ff][u][b][url=https://de.wikipedia.org/wiki/Weierstra%C3%9Fsche_%E2%84%98-Funktion]Weierstrass ℘-function[/url][/b][/u][/color] ; [math]\hookrightarrow[/math] [url=https://de.wikipedia.org/wiki/Jacobische_elliptische_Funktion][color=#0000ff][u][b]Jacobian elliptic functions[/b][/u][/color][/url] (wikipedia).[br][/size][size=85][br]The position of the [color=#00ff00][i][b]foci[/b][/i][/color] is characteristic for the solutions of the [color=#9900ff][i][b]elliptic differential equations[/b][/i][/color] of the above type.[br]If these are different, their position is determined by their [i][b]cross ratio[/b][/i] [math]d:=\frac{f_1-f_3}{f_2-f_3}\cdot\frac{f_2-f_4}{f_1-f_4}[/math].[br]Regardless of the order, the [i][b]absolute invariant[/b][/i] is [math]J_{abs}=J\left(f_1,f_2,f_3,f_4\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]Two [color=#9900ff][i][b]elliptic functions[/b][/i][/color] whose [i][b]absolute invariants[/b][/i] agree can always be transformed by a [color=#0000ff][i][b]Möbius transformation[/b][/i][/color][br]convert into each other. The [i][b]absolute invariant[/b][/i] is an invariant of the [color=#0000ff][i][b]Möbius transformations[/b][/i][/color].[/size]

[size=85]The[color=#ff0000][i][b] circles[/b][/i][/color] of an [color=#ff0000][i][b]elliptical circle pencil[/b][/i][/color] are solution curves of the differential equation, [math]p'=\frac{\left(p-f_1\right)\cdot\left(p-f_2\right)}{f_1-f_2}[/math],[br][/size][size=85]where the [color=#00ff00][i][b]focal points[/b][/i][/color] [/size][size=85] [math]f_1,f_2\in\mathbb{C}[/math] [/size][size=85]are the base points of the [color=#ff0000][i][b]circle pencil[/b][/i][/color].[/size][size=85][br]The [color=#9900ff][i][b]differential equation[/b][/i][/color] [math]p'=\left(p-f\right)^2[/math] describes a [color=#ff0000][i][b]parabolic pencil[/b][/i][/color] of [color=#ff0000][i][b]circles[/b][/i][/color] with [math]f[/math] [/size][size=85]as the point of contact.[/size][size=85][br]Any [color=#9900ff][i][b]elliptic differential equation[/b][/i][/color] of the above type can be written as the "product" [br]of two [color=#ff0000][i][b]circle pencil differential equations[/b][/i][/color], apprehend, even in different ways, depending on the position of the [color=#00ff00][i][b]foci[/b][/i][/color].[br]The solution curves of the [/size][size=85][color=#9900ff][i][b]elliptic differential equation[/b][/i][/color][/size][size=85] are [color=#0000ff][i][b]bisectors[/b][/i][/color] of the intersecting [color=#ff0000][i][b]circles[/b][/i][/color][br]from the [color=#cc0000][b]2[/b][/color] [i][color=#ff0000][b]circle[/b][/color][/i] [color=#ff0000][i][b]pencils[/b][/i][/color] of the product.[br]This is also the case when [color=#00ff00][i][b]foci [/b][/i][/color]coincide.[br]The picture at the top left shows the [color=#9900ff][i][b]elliptical directional field[/b][/i][/color] that results from the [/size][size=85][color=#0000ff][i][b]bisectors[/b][/i][/color][/size][size=85] field [br]of the two [color=#ff0000][i][b]circle pencils[/b][/i][/color]. [br][/size][size=85][u][i][b]Hint:[/b][/i][/u] For [color=#cc0000][b]2[/b][/color] complex numbers[/size][size=85] [math]z_1=\rho_1\cdot e^{i\cdot\varphi_1},z_2=\rho_2\cdot e^{i\cdot\varphi_2}[/math] is [math]w=\sqrt{z_1\cdot z_2}=\sqrt{\rho_1\cdot\rho_2}\cdot e^{i\cdot\frac{\varphi_1+\varphi_2}{2}}[/math] [/size][size=85][color=#0000ff][i][b]angle bisector[/b][/i][/color]![/size]

[size=85]If the [b][i]absolute invariant[/i][/b] [math]J_{abs}[/math] of the [b][color=#cc0000]4[/color][/b] [b][i][color=#00ff00]focal points[/color][/i][/b] of an [b][i][color=#9900ff]elliptic differential equation[/color][/i][/b] is [b]real[/b],[br]or [/size][size=85][b][i][color=#00ff00]focal points[/color][/i][/b][/size][size=85] coincide, then for suitable [b][i][color=#38761d]confocal[/color][/i][/b] [b][i][color=#ff7700]bicircular quartics[/color][/i][/b] are[br][b][i][color=#ff00ff]solution curves[/color][/i][/b] of the [b][i][color=#9900ff]differential equation[/color][/i][/b].[br]If the [b][color=#cc0000]4[/color][/b] [/size][size=85][b][i][color=#00ff00]focal points[/color][/i][/b][/size][size=85] are different, then[br] - for [math]J_{abs}\ge0[/math] the [/size][size=85][b][i][color=#00ff00]focal points[/color][/i][/b][/size][size=85] are [b][i][color=#ff0000]concyclic[/color][/i][/b], the [b][i][color=#ff0000]quartics[/color][/i][/b] are [b]2-part[/b];[br] - for [math]J_{abs}\le0[/math] [b][color=#cc0000]2[/color][/b] of the [/size][size=85][b][i][color=#00ff00]focal points[/color][/i][/b][/size][size=85] pairs 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], the [b][i][color=#ff7700]quartics[/color][/i][/b] are [b]1-part[/b].[/size]

[size=85]If [b][color=#cc0000]2[/color][/b] of the [b][i][color=#00ff00]focal points[/color][/i][/b] coincide into one, and one transforms this to [math]\infty[/math], [br]the result is a [b][i][color=#38761d]confocal[/color][/i][/b] [b][i][color=#ff7700]midpoint cone section[/color][/i][/b].[br]If [b][color=#cc0000]3[/color][/b] [b][i][color=#00ff00]focal points[/color][/i][/b] coincide in one, one obtains [b][i][color=#38761d]confocal[/color][/i][/b] [b][i][color=#ff7700]parabolas[/color][/i][/b] with this as a [math]\infty[/math].[br][br][b][i][u][color=#cc0000]Not included above are 2 special cases:[/color][/u][/i][/b] [br][br][list][*] [b][color=#cc0000]4[/color][/b] different [b][i][color=#00ff00]focal points[/color][/i][/b] with : [math]J_{abs}=0[/math] the [color=#00ff00][b][i]f[/i][/b][/color][b][i][color=#00ff00]ocal points[/color][/i][/b] are [b][i][color=#ff0000]concyclic[/color][/i][/b] and have [b][i][color=#0000ff]harmonic position[/color][/i][/b], [br] there are [b][color=#cc0000]2[/color][/b]-part [b][i][color=#ff7700]bicircular solution curves[/color][/i][/b] and, at an angle of [b][color=#cc0000]45°[/color][/b] to them, [b][color=#cc0000]1[/color][/b]-part [b][i][color=#ff7700]bicircular solution curves[/color][/i][/b]. [b] Square[/b] case with diagonals.[/*][/list][/size][list][*][size=85] [math]J_{abs}=-1[/math] [b][i][color=#ff00ff]Hexagonal case[/color][/i][/b]: On the [b][i][color=#0000ff]Möbius sphere[/color][/i][/b], the [b][i][color=#00ff00]focal points[/color][/i][/b] can be arranged as the corners [br] of a[b][i][color=#1e84cc] regular tetrahedron[/color][/i][/b]. Through each point (apart from the [b][i][color=#00ff00]focal points[/color][/i][/b]) [br] six [b][color=#cc0000]1-[/color][/b]part [b][i][color=#ff7700]bicircular quartics[/color][/i][/b] go through as [b][i][color=#38761d]solution curves[/color][/i][/b]; intersection angle: multiples of [b][color=#cc0000]60°[/color][/b].[/size][/*][/list]

[b]W. BLASCHKE's [/b]problem: [b](1938)[br][br][/b][list][*][b] [/b][i]Determine all [/i][b][color=#ff7700][i]hexagonal webs[/i][/color][/b][i] that can be formed from [color=#cc0000]3[/color] families of [b][color=#ff0000]circular arcs[/color][/b]![/i][/*][/list][size=85]The question of all [b][i][color=#ff7700]hexagonal webs[/color][/i][/b] from [b][color=#cc0000]3[/color][/b] [b][i][color=#6d9eeb]straight line families[/color][/i][/b] had already been solved in [b]1938[/b].[br]We determined [b][i][color=#ff7700]hexagonal webs[/color][/i][/b] from [b][color=#cc0000]3[/color][/b] [b][i][color=#ff0000]pencils of circles[/color][/i][/b] tufts in [b][i]1983[/i][/b], and there have been repeated [br]contributions to this in the meantime. [br]The problem is solved for [b][i]spatial [color=#ff7700]circular arcs-webs[/color][/i][/b] ([b]3D[/b]):[br]one finds such "[b][i][color=#ff7700]3-web of circles[/color][/i][/b]" only on [b][i][color=#980000]DARBOUX cyclids[/color][/i][/b]. Single-shell [b][i][color=#ff7700]hyperboloids [/color][/i][/b]are special examples of this.[br]Not solved and obviously very difficult to solve is the question of all [/size][size=85][b][i][color=#ff7700]3-web of circles[/color][/i][/b][/size][size=85] on the [b][i][color=#0000ff]Möbius sphere[/color][/i][/b],[br]or in the [b][i][color=#0000ff]complex Möbius plane[/color][/i][/b] [math]\mathbb{C}\cup\left\{\infty\right\}[/math].[br]What is a [b][i][color=#ff7700]hexagonal web[/color][/i][/b]?[br][/size][size=50][b][color=#cc0000]3[/color][/b] sets of curves in an open area, with the following properties: - exactly one curve from each set passes through each point.[br] - 2 curves from different groups intersect in the area at exactly one point....[br]is called a [b][i][color=#ff7700]hexagonal web[/color][/i][/b] if each hexagonal figure of [b][color=#cc0000]3*3[/color][/b] curves closes at a common [b][i][color=#ff0000]point[/color][/i][/b][/size][size=50][size=85]....[/size][/size]

[size=85][b][i][u][color=#cc0000]Left:[/color][/u][/i][/b] Hommage á [b]WALTER WUNDERLICH[/b]. In 1938 [b]Walter Wunderlich[/b] examined [b][color=#cc0000]2[/color][/b]-part [b][i][color=#ff7700]bicircular quartics[/color][/i][/b] and [br]showed that these [/size][size=85][b][i][color=#ff7700]quartics[/color][/i][/b][/size][size=85] have [b][color=#cc0000]3[/color][/b] families of [b][i][color=#999999]double-touching[/color][/i][/b] [b][i][color=#ff0000]circles[/color][/i][/b], [br]from which a "[/size][size=85][i][b]besonderes Dreiecksnetz aus Kreisen[/b][/i][/size][size=85]" ("[i]a special 3-web-of-circles[/i]") can be constructed.[br]To each of these [/size][size=85][b][color=#cc0000]3[/color][/b][/size][size=85] families of [b][i][color=#ff0000]circles[/color][/i][/b] belongs a [b][i][color=#bf9000]symmetry[/color][/i][/b] [b][i][color=#ff0000]circle[/color][/i][/b]. [br]The [b][color=#cc0000]2[/color][/b]-part [b][i][color=#ff7700]bicircular quartics[/color][/i][/b] have [b][color=#cc0000]4[/color][/b] paired [b][i][color=#0000ff]orthogonal[/color][/i][/b] [b][i][color=#bf9000]symmetry[/color][/i][/b] [b][i][color=#ff0000]circles[/color][/i][/b].[br]The construction of these [b][i][color=#ff7700]3-web-of-circles[/color][/i][/b] uses the [b][i][color=#00ff00]foci[/color][/i][/b] and the associated [b][i][color=#0000ff]directic circles[/color][/i][/b].[br]The constructions are successful in finding [/size][size=85][b][i][color=#ff7700]3-web-of-circles[/color][/i][/b][/size][size=85] also in the case of midpoint [b][i][color=#ff7700]conic sections[/color][/i][/b][br]and their [b][i][color=#999999]double-touching[/color][/i][/b] [b][i][color=#ff0000]circles[/color][/i][/b], to which the [b][i][color=#444444]tangents[/color][/i][/b] also belong: möbiusgeometrically is [math]\infty[/math] a [br][b][i][color=#999999]double-counting[/color][/i][/b] [b][i][color=#00ff00]focal point[/color][/i][/b] and a [b][i][color=#ff7700]curve point[/color][/i][/b]![br]In [b]2013[/b], [b]FEDOR NILOV[/b] presented [b][i][color=#cc0000]new[/color][/i][/b] [/size][size=85][b][i][color=#ff7700]3-web-of-circles[/color][/i][/b][/size][size=85] ("NEW EXAMPLES OF HEXAGONAl WEBS OF CIRCLES"):[br]For [b][i][color=#ff7700]conic sections[/color][/i][/b], these examples include not only the [/size][size=85][b][i][color=#999999]double-touching[/color][/i][/b] [b][i][color=#ff0000]circles[/color][/i][/b][/size][size=85] but also the [b][i][color=#ff0000]pencils of circles[/color][/i][/b] [br]belonging to the [b][i][color=#00ff00]focal points[/color][/i][/b].[br]We will present a [b][i][color=#9900ff]general overview[/color][/i][/b] of [/size][size=85][b][i][color=#ff7700]3-web-of-circles[/color][/i][/b][/size][size=85] in the last chapter of this [b][i][color=#980000]geogebra-book[/color][/i][/b].[br]Included are some [b][i][color=#ff0000]circle[/color][/i][/b] [b][i][color=#ff7700]webs[/color][/i][/b] that are probably unknown so far like the [/size][size=85][b][i][color=#ff7700]3-web-of-circles[/color][/i][/b][/size][size=85] shown above on the [b][i][u][color=#cc0000]right[/color][/u][/i][/b].[br]The examples of [b]FEDOR NILOV[/b] are included as special cases.[/size]