Hi all, after this:
http://www.fractalforums.com/mandelbulb-renderings/a-new-class-of-bulb/msg9988/#msg9988And something Jos Leys pointed out to me, i.e. that for the Mandelbulb Julias because the rotations do not affect the magnitude then for the analytical distance estimate for z^p+c (Julias where p is real) because we only need the magnitude of the final derivative in the end then the running derivative can be done using just the magnitude rather than the full triplex value, i.e. to get the derivative we simply iterate:
dz = p*dz*m^(p-1)
where dz is the (both running and final) magnitude of the derivative, p is the power/degree (real) and m is the current magnitude of the main iteration.
Putting the above ideas together I came up with the following for the main iterate:
Convert the triplex (or any multi-dimensional number) to the magnitude and the relevant unit vector, if the magnitude is 0 return the constant as the new value else...
New magnitude=magnitude^p
Apply *any* function f() you like to the unit vector.
If the function used is such that the result may not maintain unity then normalise the result throwing away the magnitude - special case a zero result e.g. return 1 (real) if zero generally (or another appropriate unit vector depending on your function e.g. based on numeric or geometric limits).
Now simply multiply the new unit vector by the new magnitude and add the constant.
When doing the above then the optimisation of using just the magnitude for the analytical derivative for Julia Sets still applies for any function f using power p.
Note that using the optimisation of just using the magnitude for the derivative on the Mandelbrot versions does work to some extent but the calculations really require the full vector derivative because of the "+1" added on each iteration.
January 25, 2013, 11:59:39 AM
Dilber MC Theme by HarzeM