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Parametrize plots on the imaginary plane to a real axes plane?

Bryan Lettner
Posted 10 years ago

"For every plot generated using imaginary numbers, there exists an alternate parameterization using only real numbers."

Something I've been thinking a lot about. I assume it has a name, is well-studied, and has been proved or disproved already.

Let's use the Mandelbrot Set as an example. We could re-label the "real" and "imaginary" axes "x" and "y" if we wanted. Surely, there must be a simple parameterization x = ( something ) , y = ( something ) that generates the boundary of the set without utilizing Sqrt[-1] ?
POSTED BY: Bryan Lettner
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Somewhat more elegant would be:

plt = ContourPlot[{Abs[x + I y] == 2,    Abs[x + (x + I y)^2 + I y] == 2}, {x, -2, 2}, {y, -2, 2}]
curve = 2;
ind = Cases[List @@ plt, Line[x_] -> x, Infinity][[curve]];
ppp = List @@ Cases[List @@ plt, GraphicsComplex[x_, y__] -> x][[1]];
points = ppp[[#]] & /@ ind;
plt2 = ListPlot[points]
Show[plt2, plt]
POSTED BY: Hans Dolhaine

I am pretty sure there is a more elegant solution to your problem. But try this

plt = ContourPlot[{Abs[x + I y] == 2,  Abs[x + (x + I y)^2 + I y] == 2}, {x, -2, 2}, {y, -2, 2}]

curve is the number of the curve which you want to extract:

curve = 2;
ind = Flatten[List @@ plt[[1, 2, 1, curve + 2, 2]]];

Now get the points you want to have:

points = List @@ plt[[1, 1]][[#]] & /@ ind

and look

    plt2=ListPlot[points]
    Show[plt2, plt]
POSTED BY: Hans Dolhaine

You can get the implicit relation as below.

ee = x + (x + I y)^2 + I y;
eesq = ComplexExpand[Expand[ee*Conjugate[ee]]]

(* Out[170]= x^2 + 2 x^3 + x^4 + y^2 + 2 x y^2 + 2 x^2 y^2 + y^4 *)

A parametric form is a different matter though. Not sure what might work for that.
POSTED BY: Daniel Lichtblau

Dear @Bryan Lettner does your question have any relation to Wolfram Technologies?

This forum permits only subjects related to Wolfram Technologies. Post elsewhere for other subjects or make clear the connection to Wolfram Technologies. Please read the guidelines.
POSTED BY: EDITORIAL BOARD
Bryan Lettner
Bryan Lettner
Posted 10 years ago
POSTED BY: Bryan Lettner
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