u/ColorPlaysLmao

I was messing around with a cubic formula I made in Desmos and i noticed that if i create an implicit function that depends on the expression real(f(x+yi)) = 0 (where f(x) is the cubic function with complex coefficients) it forms a sort of "3-way hyperbola", for lack of a better name

I am relatively sure that this is some sort of 3-way hyperbola, because in the case of the quadratic, the same implicit function results in an actual standard hyperbola,

and I assume that because a quadratic has 2 roots, its a standard "2-way" hyperbola, and the cubic having 3 roots creates a 3-way hyperbola. and this would extend to quartic and higher polynomials, i.e. the implicit function real(f(x+yi))=0 for f(x) being any n-degree polynomial creates a 'n-way hyperbola'

Cubic and Quadratic Formula | Desmos

All my attempts to describe the curve with implicit functions haven't gone anywhere, I'm kind of interested in further researching this curve and I can't find any references to it online, so if anyone knows of any references of it, tell me pleas! thank u!

Orginally, I just ported the cubic formula directly into desmos, and it seemed to work fine for real coefficients as a,b,c and d.

However, I then noticed a weird issue that emerged when i started introducing complex coefficients, The "roots" it showed would instantaneously jump around the place, which was not meant to happen, as it would imply a sort of fundamental discontinuity within the structure of the polynomial.

So then I tried inputting the value it said was a "root" into the original function, and it was just not a root at all, for example, one of the "roots" it showed me actually corresponded to a value of almost 1000.

While I am not certain, I believe the reason this was happening is because in practice, the cubic formula has many different 'cases' in which you have to apply the formula differently, for example: if you have a perfect cube, and you change the value for d, you must change the sign of the square root within one of the values in the cubic formula after all the roots cross through eachother (Corresponding to the triple root of a perfect cube), and this must be done to get accurate solutions.

But in principal, how I did it mainly revolves around getting one principal root (that is actually a root) using list inequalities and then using a generalised synthetic division formula that takes in one root and the coefficients of the polynomial and automatically gives you the other two roots (you can see this process using the custom variable 'H' in the graph, if you set it equal to a root it outputs the two other roots

for quadratics, the quadratic formula works easily. no issues there.

THESE GRAPHS INCLUDE:

2D DOMAIN COLORING:

- Setting to swap whether 0 is white or black

- Slider for the vibrancy of the function

- Slider to interpolate between all values of discontinuous functions (equivalent to multiplying by e^iθ)

3D DOMAING COLORING:

- Setting to swap whether 0 is white or black

- Slider for the vibrancy of the function

- Slider to interpolate between all values of discontinuous functions (equivalent to multiplying by e^iθ)

- Riemann Surface generator up to 6 different roots of unity

- Implicit domain coloring

- Option to switch whether the z axis represents the real or imaginary component

Check optimization settings, sometimes you have to tweak around with them to get the grid working for a graph

theres lowkey so much to explore in this graph

Im tempted to add things like versine, coversine, exsecant, all those other ones but idk if it would work well in terms of geometric visualization

Desmos doesnt support them, so i made them work myself..

its very laggy