[br][table][tr][td][url=https://www.geogebra.org/m/nzfg796n#material/hz2rbzcj][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]

[br][size=85][color=#cc0000][b][size=100]move[/size][/b][/color] [color=#980000][b]a, b, c[/b][/color]; [color=#274E13][size=100][b]change[/b][/size][/color] [math]\mathbf{n}[/math], [math]\mathbf{ty_{max}}[/math], [math]\mathbf{tx_{max}}[/math] [br][br]The complex [b][i][color=#0000ff]exponential function[/color][/i][/b] forms [br] the [b][i][color=#ff0000]parabolic pencil of circles[/color][/i][/b], which consists of the axis-parallel [b][i][color=#ff0000]straight lines[/color][/i][/b], [br]to the [b][i][color=#ff0000]elliptic-hyperbolic pencils of circles[/color][/i][/b] consisting of the [b][i][color=#ff0000]straight rays[/color][/i][/b] emanating from the origin [br] and the [b][i][color=#ff0000]concentric circles[/color][/i][/b].[br]The [b][i][color=#0000ff]exponential function[/color][/i][/b] is simply periodic: the [b][i][color=#ff0000]parallels[/color][/i][/b] to the [math]y[/math]-axis are mapped onto the [b][i][color=#ff0000]concentric circles[/color][/i][/b]:[br] change the parameter interval using [math]\mathbf{ty_{max}}[/math].[br]The [b][i][color=#ff0000]pencil of parallels[/color][/i][/b] which intersect the axis parallels at a constant angle give [br]parallel [b][i][color=#38761d]loxodromes[/color][/i][/b] under the exponential function: these are the curves which intersect the origin rays [br]at a constant angle: [b][i][color=#38761d]logarithmic spirals[/color][/i][/b].[/size]