"Infinitizing" the poles of the Juliabulb

[msltoe]

In the original Juliabulb, the poles are a sore spot. What I came up recently is a way to make them more appealing. I don't know how to describe it with words except to call it "infinitizing". This works with (a,b,c) seeds but the Mandelbulb doesn't seem to work out quite right. Also the infinitizer I use isn't quite perfect. Perhaps, others can figure out a better formula.

   r=x*x+y*y+z*z;
   theta = atan2(y,x);
   phi = atan2(z,sqrt(r-z*z));
   w=8/(sqrt(1-(phi*phi)/(pi*pi*0.5*0.5))); //infinitizer
   r=pow(r,w/2);
   phi = w*phi;
   theta=8*theta;
   x=r*cos(theta)*cos(phi);
   y=r*sin(theta)*cos(phi);
   z=-r*sin(phi);

   Image #1: { x = x-0.8;y=y-0;z=z+0;} with power 8.
   Image #2: ( x=x-0.7;y=y-0.5 (?);z=z+0;} with power 2.

[hobold]

Is that (the analogy of) a Riemann sphere with two poles?

[msltoe]

It does look a lot like that. What's interesting is that it's actually accelerating the scaling of phi and r as it reaches the poles vs. slowing down "phi" in your monopolar formula.

i found a better infinitizer:
scale phi by order/(abs(cos(phi)))^(1/4)
raise r to 0.5*order/sqrt(cos(phi));

I'm still puzzled why the M-sets or c != 0 seeds don't look right.

-mike

[hobold]

Riemann's original mapping is unique in being conformal, a quality that my distorted monopolar mapping did not preserve. Maybe your mapping could be made (mostly) conformal by stitching two Riemann hemispheres together ... ?

[msltoe]

The Riemann sphere is a smooth mapping of the complex plane on the surface of the sphere, the problem is that towards the pole or poles (if we were to do stitching), we're moving to parts of the complex plane that are far away from the origin and zero for the M-set, because they escape to inifinity. Still, the qualitative similarity is eerie.

Even though I've managed to normalize the features of the bulbs as they proceed towards the poles, I haven't removed the non-conformalness between the bulbs. (i.e., order * phi approaching pi/2 or -pi/2).

two iterations of original 8th-order juliabulb and infinitized version:

[twinbee]

Great stuff msltoe - I've thought that achieving the sunflower effect (tinier and tiner buds towards the pole) like you have would help the rest as well, though apparently not.

I'd like to see if this can be done with the standard Mandelbulb as well as the Juliabulb. It seems like you're edging towards the grail, at least hopefully!

[msltoe]

twinbee: Thanks for the encouragement. It seems I'm trying to do the impossible: making angle doubling conformal or near-conformal.

The "m-set" (a,b,0) and a fuzzy Julia with some extra operators:

[knighty]

Awesome idea!
I don't fully understand it (yet) but maybe the inifinitizer should depend on the seeds? Just an idea. Not sure.
About conformality: If I understand correctly the concept, it's related the the condition number of the jacobian of the map. That is, the ratio of the biggest to the smallest singular values of that jacobian (let's call it alpha). I know, thanks to Tglad's explanations, that In n-dimension (n>2) , only reflexions, translations, uniform scaling and inversion (and their combinations, including rotations that are the combination of two reflexions) are conformal. Now what about near conformality or alpha-conformality?
The question is: How to build a map that is alpha-conformal. Which jacobian's condition number is less than a given positive number everywhere?
More importantly: Is there an alpha-conformal map that when iterated (infinitely?) is alpha'-conformal? If the answer is yes, how to buld it?
(Sorry for the mathematical inaccuracies and the bad english. I hope it's clear enought  )

[msltoe]

knighty: You may be right about the seeds affecting the formula for the infinitizer. Honestly, your handle on the math is better than mine. I used my intuition and some trial and error to achieve the desired effect. The idea I worked with is that spherical coordinates are orthogonal with a diagonal metric tensor. One of the terms in the tensor is cos(phi), if we're using the z=rsin(phi) convention. So I may be able to justify the 1/cos(phi) function for normalization?

My dream of conformality in spherical coordinates (with r'=log[r]) stems from the fact that spherical coordinates are orthogonal and the transform we want to apply is just multiplying by 2 for each coordinate. So then what is left is normalizing the metric.
 

[knighty]

All you can hope for is a near conformal mapping. The infinitizer would do a very good job.
Reminds me my attempt at solving the problem of overestimations in mandelbulb DE that occure near the "flower" shapes. The formula I've found depend on the seed cos(phi) and cos(p*phi) (btw: thank you hobold). It's an non "perfect" empirical formula and all my attempts to analyse the problem failed . The fact is that we NEED the HELP! of a mathematician.