An example of finding explicit equations for curves using complex functions that make up an implicitly defined cubic curve
Statement of the problem:
The curve is given in implicit form as g(x, y) = 0, for example a circle as y² + x² = 1.
Find:
the explicit form of the equations y=f(x) for each of the k branches of the curve, i.e. {fᵢ(x)}, where i=1..k. For a circle, for example, f₁(x) = sqrt(1-x²) for y > 0, and f₂(x) = − sqrt(1-x²) for y < 0. If f(x, y) is a quadratic function with respect to variable y, GeoGebra can easily find the roots in symbolic form, y₁(x) and y₂(x).
For polynomials of the 3rd and 4th degree, knowing the existing rigorous solutions of their equations in symbolic form, one can find the equations of the branches {fᵢ(x)} of the corresponding plane curves using complex functions.
Earlier, I considered biquadratic equations for the Trifolium curve and the Cartesian oval. These examples illustrate the application of complex functions. In these cases, it is possible to find the equations of the branches of curves using both real functions and complex functions, from the real part of the functions of which only those sections are left where the corresponding complex part is missing, i.e. Im(x)=0.
This applet considers an example from mathstackexchange: Branches of implicitly defined cubic curves for an implicitly defined cubic curve a y³ + b y² + c y + d(x) = 0 where d(x)=-5 x³ + x² + 45x - c0, for which I found explicit equations of the curves of its branches using complex functions.
Images of three examples are in the applet.