Non-uniform "power" and the full derivative

[David Makin]

Hi all, am just implimenting the White/Nylander Mandelbulb (and others) as class-based plug-ins for a new 3D+ Ultra Fractal formula.
I want to make the formulas for the various Mandelbulbs as versatile as possible and to this end would like to know if there's a sensible answer to the following:

Taking the White/Nylander as an example, if we split the power into 3 different values, powermagnitude, powerphi and powertheta such that in the calculations the magnitude is raised to powermagnitude, the phi angle is scaled by powerphi and the theta angle by powertheta then we have a system with much more variety than the standard Mandelbulb and raising a given triplex to a (triple) power is straightforward.

My question is how does this affect calculation of the full derivative value  - normally power*dz*z^(power-1) ?

I would assume that calculating q=z^(power-1) would be done by subtracting one from powermagnitude and powerphi and powertheta (or just from powermagnitude???) and the constant "power" would be powermagnitude - but I can't work out the "correct" way to do "dz*q" using such a triplex of powers.

I realise that doing it without the derivative (i.e. using delta DE) or using the magnitude-only version of the derivative are both possible but I'd like to include the full derivative version too.

[David Makin]

Just to add I was thinking that using 3 separate powers then a multiplication normally:

phi = phi1 + phi2
th = th1 + th2
mag = mag1*mag2

Is instead:

phi = (phi1+phi2)*powerphi/powermagnitude
th = (th1+th2)*powertheta/powermagnitude
mag = mag1*mag2

[kram1032]

Hmm, well, looking at complex numbers, you'd substract one from the real part, not quite from the magnitude or the angle... So I guess, you'd have to do the same for triplex, substracting 1 from the real part?

[David Makin]

Hmm, well, looking at complex numbers, you'd substract one from the real part, not quite from the magnitude or the angle... So I guess, you'd have to do the same for triplex, substracting 1 from the real part?

Hmmmm - you mean you think treating powermagnitude, powerphi and powertheta as a triplex in polar form and converting it to cartesian, subtracting 1 and then converting back would do it ?
Effectively I think that gives us triplex^triplex maths if it works ?

Based on that then I assume a multiplication would just be done normally i.e. changing the powers doesn't alter the natural multiplication rules in the way I was thinking ?

[kram1032]

Really, I'm no expert and triplex algebra behaves strangely anyways but d/dx x^(complex power)  = complex power * x^(complex power - 1)
I don't really see, why that would change for triplex... However, only trying it out would show, right?

[trafassel]

For this video i replaced the angles in the Mandelbulb formula (White variant) with the 3 angles you suggested. In the corresponding Juliabulb  there are regions which not fit.

<a href="http://www.youtube.com/v/4l1Fk0yYObM&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/4l1Fk0yYObM&rel=1&fs=1&hd=1</a>

[kram1032]

hmmm

well, as x^n is generally different for triplex than x*x*.....*x*x (n times), you might have to special-case it in a way. Most likely, that's not quite a trivial problem.

In most cases, the renders seemed to work fine
That shader is great when it's animated, btw