[size=85] Solving an implicitly defined plane curve means splitting it into separate "curve sections" for which [b][i]explicit[/i][/b] expressions of functional dependencies can be written. These may take the form of analytical function definitions, y = f(x), or parametrically defined curves where the x-coordinate coincides with the parameter, x = t.[br] In case of the [b][url=https://en.wikipedia.org/wiki/Cartesian_oval]Cartesian oval[/url] [/b]where an implicit equation of a plane curve is given in the form of a [i]biquadratic equation[/i] in the variable [b]y[/b]. Using its [url=https://eqworld.ipmnet.ru/en/solutions/ae/ae0104.pdf]root formulas[/url],[i] 4 explicit functional dependencies[/i] of the real variable [b]x[/b] are found that make up the plane curve under consideration. However, this method is not always applicable.[br] Like the previous [url=https://www.geogebra.org/m/kwdbdznt]applet[/url], [b]this applet[/b] illustrates how to use of the [i]method of parametrically defined complex functions[/i] to solve [b]the same implicit [i]biquadratic equation[/i][/b] in [b][color=#1e84cc]y[/color][/b]: [color=#1e84cc][b]eq[/b][/color]: ([b]x[/b]² + [b]y[/b]² - 2a [b]x[/b])² - b² ([b]x[/b]² + [b]y[/b]²) - c=0. Alternatively, it can be expanded into powers of [b]y[/b]: [b]y[/b][sup]4[/sup] + ( -4 a [b]x[/b] + 2 [b]x[/b][sup]2[/sup] - b[sup]2[/sup]) * [b]y[/b][sup]2[/sup] +(4 a[sup]2[/sup] [b]x[/b][sup]2[/sup]-4 a [b]x[/b][sup]3[/sup]+[b]x[/b][sup]4[/sup]-[b]x[/b][sup]2[/sup] b[sup]2[/sup]-c)=0. Using the new variable assignments: k1:=a; k2:=b; k3:=c and the function assignments b([b]x[/b])=(-4 k1 [b]x[/b] + 2 [b]x[/b]² - k2²); c([b]x[/b])= 4 k1² [b]x[/b]² - 4 k1 [b]x[/b]³ + [b]x[/b]⁴ - [b]x[/b]² k2² - k3 we can rewrite the equation as [b]y[/b][sup]4[/sup] + b([b]x[/b]) [b]y[/b][sup]2[/sup] + c([b]x[/b]) = 0. With the parameter [b]a[/b] before [b]y[/b]⁴, we have a*[b]y[/b]⁴+b([b]x[/b])*[b]y[/b]²+c([b]x[/b])=0. [br] In the complex plane the variable [b][color=#1e84cc]x[/color][/b]→[b][color=#1e84cc]z[/color],[/b] [b][color=#1e84cc]y[/color][/b]→[b][color=#1e84cc]f(z)[/color][/b]. The root formulas used can be easily extended to the complex plane in a [url=https://www.reddit.com/r/geogebra/comments/1k35f1x/the_calculation_of_complex_numbers_in_algebra_is/]certain way[/url]. Roots: complex functions [b][color=#ff00ff]g1(z)[/color][/b],[color=#ff7700][b] g2(z)[/b][/color], [b][color=#0000ff]g3(z)[/color][/b] and [b][color=#38761d]g4(z)[/color][/b] are solutions of the corresponding complex equation.[br] *Images made with this applet can be seen in the [url=https://www.geogebra.org/m/h7p4k3np]applet[/url].[/size]

[url=https://www.mathouriste.eu/Maxwell/Maxwell_Ovales.pdf]https://www.mathouriste.eu/Maxwell/Maxwell_Ovales.pdf[/url]