Fractal in 3D associated with z^3-1

Hello guys,
Recently, I’ve been interested in doing 3D Fractals. After several attempts to code such a fractal, I come for help.
I have tried to use the following method : you have a fixe point P=(x,y,z) with z>0, for points of the form A=(x,y,w=0), you calculate w’=-k/u^3 for some k (e.g. 0.75), and then, if -k/u^3 is not near enough the altitude w you shift A on the line (AP) (and the direction, depending on w>w’ or w’>w) and you iterat the process until w and w’ are near enough.
To accelerate the computing of the points, I have used this reference http://www.fractalforums.com/mathematics/convergent-distance-estimation-t1566/ instead of using a constant shift distance.
However, I’m not familiar with fractals and thus I am wondering :

-How much iteration should the process of getting from A=(x,y,w=0) a point near enough the surface take ? I have been stopping the calculation at more than 1000 iterations with a precision of 0.0001 for w-w’. Do you think I should lower (at least in a first time) the precision ?

  • For the distance estimation, it seems the relation is K=ln(|zn+1-zn|)|zn+1-zn|/|dzn+1-dzn| with dzn+1=dznP(zn)P’’(zn)/(P’(zn)^2) and zn+1 and zn are near enough. However, i don’t understand what value of dz0 should be taken (right now i have taken z1-z0), especially since it matters a lot in the estimation of the distance and so in my algorithm. Also should the shift be done of exactly the distance, or some constante times the distance (e.g. 0.9*distance) ?

-Do you know any cool ressources to learn that I could use that use this method ?
Thanks in advance !

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