graphics - Rendering Fractals: The Mobius Transformation and The Newtonian Basin -

graphics - Rendering Fractals: The Mobius Transformation and The Newtonian Basin -

i understand how render (two dimensional) "escape time group" fractals (julia , mandelbrot), can't seem mobius transformation or newton basin rendered.

i'm trying render them using same method (by recursively using polynomial equation on each pixel 'n' times), have feeling these fractals rendered using totally different methods. mobius 'transformation' implies image must exist, , transformed produce geometry, , newton basin seems plot each point, not points fall set.

how these fractals graphed? graphed using same iterative methods julia , mandelbrot?

equations i'm using:

julia: zn+1 = zn^2 + c

where z complex number representing pixel, , c complex constant (correct).

mandelbrot: cn+1 = cn^2 + z

where z complex number representing pixel, , c complex number (0, 0), , compounded each step (the reverse of julia, correct).

newton basin: zn+1 = zn - (zn^x - a) / (zn^y - a)

where z complex number representing pixel, x , y exponents of various degrees, , complex constant (incorrect - creating centered, 8 legged 'line star').

mobius transformation: zn+1 = (azn + b) / (czn + d)

where z complex number representing pixel, , a, b, c, , d complex constants (incorrect, falls set).

so how newton basin , mobius transformation plotted on complex plane?

update: mobius transformations that; transformations.

"every möbius transformation composition of translations, rotations, zooms (dilations) , inversions."

to perform mobius transformation, shape, picture, smear, etc. must nowadays in order transform it.

now how newton basins?

update 2: math wrong newton basin. denominator @ end of equation (supposed be) derivative of original function. function can understood studying 'newtonroot.m' mit matlab source-code. search engine can find quite easily. i'm still @ loss how graph on complex plane, though...

newton basin:

f(x) = x - f(x) / f'(x)

in mandelbrot , julia sets terminate inner loop if exceeds threshold measurement how fast orbit "reaches" infinity

if(|z| > 4) { stop }

for newton fractals other way round: since newton method converging towards value interested how fast reaches limit, can done checking when difference of 2 consecutive values drops below value (usually 10^-9 value)

if(|z[n] - z[n-1]| < epsilon) { stop }

graphics geometry fractals