Cardeoid transform

Anybody know how to do this?

This might help you: https://en.wikibooks.org/wiki/Fractals/mandelbrot-graphics

I think the code isn't public, but I'm sure Claude would be willing to share or at least talk about it.

Interesting link

Seams he had the same idea i mentioned in
https://www.ultrafractal.com/forum/index.php?u=/topic/925/rotation-in-formula/1#post-3001

https://mathr.co.uk/blog/2013-12-16_stretching_cusps.html

.

I ask my self if he found a solution for the bulb size.
It could be that Feigenbaums bifurcation could play a role there...

He's active in the Fractal Chats discord, feel free to join. Lots of happy nerd stuff going on there. Here's the invite link in case you are interested.

https://discord.gg/DrxFcsRA

Everyone else is welcome too, ofc!

Tanks much!
(I am not a mathematician, not sure if they want me there...)

I found in his codes a solution for the bulb size correction, in case you are still interested.
https://github.com/adammaj1/Mandelbrot-book-book

(There is probable a more elegant way to calculate the bulb size, and my codes are ugly as hell, but for now i solve a problem after 35 years i tried the first time...)

The code is public, it just took some search effort.
https://mathr.co.uk/source-codes/mandelbrot-numerics.pdf
https://mathr.co.uk/source-codes/mandelbrot-graphics.pdf
https://mathr.co.uk/source-codes/mandelbrot-web.pdf
etc (insert name-of-include in the pdf filename)

huu, i was searching around the website but i missed the pdfs, tanks!

In the end, mathr.co.uk didn't help me one iota.

How I tackled this is first by taking the transform pdf.uxf/pgd_UnrollCardioid

It has a very simple formula (1 - sqr((0,1)*(4/(#pixel-(0,2))-(0,1))-1))/4

This is only one-way, but I experimented with every value in it a bit, and decided it was easiest to leave it, especially since WolframAlpha managed to simplify it to something as small as 1/sqr(#pixel) + 1/4. Notice here that 1/4 is the center of the cardioid, so I replaced it with a param.

Then the search for the inverse of that formula started, and it was just a matter of reversing the operations, and taking into account that a square root has two answers (this bit me at first, but visually stacking the forward and reverse transforms made clear the domain split was on the X axis): ±sqrt(1/(#pixel - @center))

Finally, rotation is simply horizontal shifting of the unrolled cardioid, so I added that as a conditional variable, only visible when the transform is in 'reverse' mode. This made clear that either (or both) transforms were inverted along the Y axis (in mandelbrot space) or X axis (in unrolled space). Actually it was both, and had to add a temp variable that inverted the shift if it was in the upper-right or lower-left quadrant.

unroll-test {
::J36s9jn29OVzuxJMQ47Ix7glz5l1wCLZTlliSbO2LdfAq8iHA3Ysp2DJ72n+OQJbialq6pyJ
  br57n5bYaDqGUZvLNhxQDaBJfyF8W7GEiIn9iRj9ybFCWPY66xljW1FIEl5s2gaAiyyaW7YU
  WNTSTA0GMK5/wPs50H2vNf/2CRRJPNZB2iSNqR04dS+Dqmn6C+Jnmz8jqGDeRmTKMAYvXLHm
  soZUFjpJDqxRjr7XwBHCB5ms6qtUtqOnMPb3/83SvGUuYrPMElFpJXvtQfrxCOqxk8hvpZjB
  zzKEymO3yZkwhLS+nVONYPF84HVBtxb0fxjq5WiPTw4XXdoIrgcYmgeJAPTZGIvAx5beUyF3
  fj44xdUXc/Nll57F8/rOx5XNC9caysmTW1isDqzmZQFVU9QopHaeSSTpu+35ojIJNpZ2U7wi
  W/hj4E/RUFIF2OL8o/lZSXOfSZs+JUmXcbaixFNa4u3mLUZOvDYRv1ollFHKPIyr3T2kwctW
  NQ4ofY4lLBZlYNGfk/GNxvHYBYEUIx4fJSbsXj0HAka+PZIr7agHjoZQh+waeu2EzRWXQpNE
  oFzEH8eaThC1w6Ag2PMUUcWKYNernmBr33d4w6L57rrrLypFn1f+/NqeF/rrGFVUp/UfQxy0
}

mjd_private.uxf:MandelbrotCardioidRotation {
; Transforms between Mandelbrot's cardioid coordinates <-> cartesian coordinates.
;
; 20240615 Mark jeronimus <mark.jeronimus@gmail.com>
::wtUIvhn2NGVyONMMQ07Ix/wQ4SCQaTRiLgKEpeGhEifgpxjLWyxOMxg4zH7k4sIyhezzybZe
  2xopVa56Hv8CAUS4qSm+h4WKUDw1NqfJNsH2tt9LOtvMDuFKrIjj4wWkOutUbRHwWXfpnOmQ
  dE1Nqa80IFPvHK6XDCI8SUOCcix4s8ZDNClcVBWw5oz9+2lub7wSQe06ZhzYVJHhmfmYjOa4
  hgk43aXXk6UON5pK5V0II9xw5cAZhyqEwHx0PJsaDyYNMlrAUhNOl1EgfordSf/BB89TL2c/
  D3BFbKygnejVnUGwKB3nEMTQJjVOUPYxOdmEcx/9MFfvv/L/TTJOkVrwVMSm78416HihmJxU
  DPqJR3H/cLMJk/qWKzfAvg7w5A==
}

mjd_private.ucl:BetterDistanceEstimator(OUTSIDE) {
;
; Written by Damien M. Jones
; Improved by Mark Jeronimus
;
::EK4X7in21AVwqNMMM07By/gI5SaXX7yOOYwYsub72OOKomo0YwxKzWm12v+pY3a8B/kevney
  GnRepsAgOea2Snh+rwrQT7G4pVlFWmnTdTVfbm/j8wao+6hmM4x2VKW7+A0WWMYcoNxv246V
  zUny8Wb5TNd4xQT91VLSu/G2lf2rlP00uLTfhR4Xv0UPhncaQ6pBMazJVMilUrrenERzzHmg
  guOC2HEzEKsvah2M6xJIZ3CU3QcWMsbR5+zzsjcSVuzN31OPv9pcpRjbBX99oJAhROa7hjEE
  IBEG0p0NCyIB0NnAeIh/JL/+Zg9TRLCX4IgeCiBj70W4T2Dfh6Xk9onlNqSdK6NGio1eRjRK
  ZkrPtFlF/znO9JL=
}

NB: My BetterDistanceEstimator isn't related to this transformation. I'm just animating the density along with the rotation.

Ohh, Tans much! i could not get it done until now.

(You have to be a skilled mathematician to get attention i think.)

I had my time with the distance coloring too. Here is my variant. The hard one should not need to much correction over rotation.
(i translate it from CP, but should work.)

mjd_private.ucl:BetterDistanceEstimatorNew(OUTSIDE) {

init:
  complex dz = (1, 0)
loop:
;  dz = @power * #z^(@power-1) * dz + 1
  dz = (dz * #z ^ (@power - 1) * @power) + 0

final:
;  #index = (@power*log(cabs(#z)) * cabs(#z) / cabs(dz))^(1/@power) * sqrt(#magn)

float value
if (@distance == "Soft")
    value = (real(@power) * log(cabs(#z)) * cabs(#z) / cabs(dz) * #magn) ^ (1 / real(@power))
    value = sqrt(value) * @density
else
    dz = @power * #z * log(cabs(#z)) / dz
    value = cabs(dz) * #magn * #width * @density  
    if (value >= 0.98) ;Limit Correction
        value = 0.98
    endif
endif

  #index = value


default:
  title = "Better Distance Estimator New"
  param power
    caption = "Exponent"
    default = 2.0
    hint = "This should be set to match the exponent of the \
            formula you are using. For Mandelbrot, this is usually 2."
  endparam
  int param distance
    caption = "Distance"
    default = 1
    enum    = "Soft" "Hard"
  endparam
  float param density
    caption = "Density"
    default = 1.0
  endparam
}

I always try to expand the formulas so i can use it for higher powers. In case you are interested, here some work in progress.
In this case higher powers not work jet, the formula must be made fit for Epicycloid.
But, center is calculated, sign simplified, rotation slowed down, formula combined.

complex center = (1 / @power) ^ (@power / (@power - 1))   ; Center shift for any power

if (@rollBack)

    float r = sqrt(abs(@rot))
    float sig = 1
    if (imag(c) < 0), sig = -sig ,endif
    if (@rot < 0), sig = -sig ,endif

    c = (1 / (c - center)) ^ (1 / @power) + (r * sig)   ; Roll up to Cardioid

endif

c = (1 / c^@power) + center  ; Unroll to Line