[size=85][size=50][b][color=#ff7700][right]06.08.2023[/right][/color][/b][/size]Only for [math]\delta\in\left\{-1,0,+1\right\}[/math]:[br][b][i][color=#9900ff]implicit equation[/color][/i][/b] of the curve[br][/size][list][*][size=85][math]\left(x^2+y^2\right)^2-2\cdot A_x\cdot x^2-2\cdot B_y\cdot y^2+\delta=0[/math] this is a [b][i][color=#ff7700]bicircular quartic[/color][/i][/b][/size][/*][*][size=85]with [math]A_x=\kappa\left(s\right)[/math] and [math]B_y=\frac{\kappa\left(f\right)\cdot\kappa\left(s\right)-\delta}{\kappa\left(f\right)-\kappa\left(s\right)}[/math]; with the [b][i][color=#76a5af]real function[/color][/i][/b] [math]\kappa\left(u\right)=\frac{1}{2}\cdot\left(u^2+\frac{\delta}{u^2}\right)[/math].[br][/size][/*][/list][size=85][b][i][color=#9900ff]Elliptic differential equation[/color][/i][/b] for a [b][i][color=#ff00ff]elliptic function[/color][/i][/b] [math]z\mapsto g\left(z\right)[/math]:[br][/size][list][*][size=85][math]\left(g'\right)^2=c\cdot\left(g^2-f^2\right)\cdot\left(g^2-\frac{\delta}{f^2}\right)[/math] for a suitable [math]c\in\mathbb{C}[/math].[br][/size][/*][/list][size=85][b][i][color=#ff00ff]Solution curves[/color][/i][/b] are [b][i][color=#6aa84f]confocal[/color][/i][/b] [b][i][color=#ff7700]bicirkular quartics[/color][/i][/b].[br][br]Link [math]\hookrightarrow[/math] [url=https://www.geogebra.org/m/y9cj4aqt][b][i][u][color=#0000ff]elliptic functions & pencils of circles & bicircular quartics ...[/color][/u][/i][/b][/url][/size]