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Author Topic:  _Asdam_transform in M3d  (Read 156 times)

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Offline dankje

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_Asdam_transform in M3d
« on: March 09, 2022, 03:42:55 AM »
Alef - Does this mean I can use z = z + factor*z/|z| in the same way that I use z = z^2 + c in my simple program?

And is factor just a Real number that has to be tweaked to get the best image result?



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Offline Alef

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Re: _Asdam_transform in M3d
« Reply #1 on: March 09, 2022, 05:48:26 PM »
No, this is additional transformation based on unit vector. It could be placed before or after fractal formula. It woun't generate fractal alone. Kind of:
Z=Z^2 + C
Z = Z + factor*z/cabs(z)

imaginary factor will rotate fractal, factor = -0.4 will generate large mandelbrot with rings and ringed julia sets, factor = 0.1 will shrink mandelbrot set. It is kind of like air in balloon, just that - means more air. But it needs just a small amount.

Alsou z could be multiplied by unit vector. Like:
Z=Z^2 + C
Z= Z^2/cabs(Z)

Here |z| = cabs(z)= real modulus function, not modulus squared like in Ultra Fractal of Fractint.

http://www.fractalforums.com/new-theories-and-research/is-there-anything-novel-left-to-do-in-m-like-escape-time-fractals-in-2d/msg38948/#newinit 

Unit vector:
https://en.wikipedia.org/wiki/Unit_vector 



Alsou modulus function could be calculated in different way. https://en.wikipedia.org/wiki/Lp_space#The_p-norm_in_finite_dimensions
Say Lp=1 modulus
abs(x) + abs(y)
Lp=2 modulus (cabs()  )
sqrt ( x*x + y*y )
Lp=4 modulus
( X^4 +  y^4 )^(0.25)
generalised:
Pnorm = ( abs(x)^@power + abs(y)^@power )^(1/@power)

So that Lp=1 will have rombs where Lp=2 have circles and Lp=4 would have pillows
by Edgar Malinovsky aka Edgars Malinovskis.

Offline Alef

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Re: _Asdam_transform in M3d
« Reply #2 on: March 09, 2022, 05:51:49 PM »
But a modification z=abs(z) could be more easy start. Say z=abs(z^2+c)  or z=abs(z^2)+c. One was a 'burning ship' another was 'buffalo'. https://en.wikipedia.org/wiki/Burning_Ship_fractal

Or more harder:
;from complex x=real(z)  y=imag(z)
x = x + abs( x - @foldX) - abs( x + @foldX)
y = y + abs( y - @foldY) - abs( y + @foldY)
; make complex z=x + 1i*y

Z = Z^2 + C

abs() is absolute value, cabs() is complex modulus. @ in Ultra Fractal marks user changable variable.
« Last Edit: March 09, 2022, 06:06:17 PM by Alef »

Offline dankje

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Re: _Asdam_transform in M3d
« Reply #3 on: March 10, 2022, 05:40:52 AM »
Based on previous talks, I wrote this chart which represents the state of my code.  I'm trying to codify the steps needed in determining pixel colors based on user's "palette" choice (eg. banded, 1 color gradient, 2 color gradient plus type of index fn, exponential smoothing, etc.).  Is my chart at all correct?

You mentioned these functions:
Z = Z + factor*z/cabs(z)
Z= Z^2/cabs(Z)
Pnorm ( Z?) = (abs(x)^@power + abs(y)^@power )^(1/@power)

Are they what I am calling the Index Function?

Thanks for your help.

Offline Alef

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Re: _Asdam_transform in M3d
« Reply #4 on: March 10, 2022, 05:35:10 PM »
No, it is iterated together with z=z^2+c, a fractal function.
In M3D there are 6 slots where fractal formula or transformation could be placed, and then iterated.


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