I didn't mean to order you to get to work. :-) I just wanted these experiments to be documented well.

LOL

I said so just because I was about to try it.

My Provisional conclusion is that it doesn't make a big difference because the mandelbulb (I'm talking about power 8 mandelbulb) surface is near the radius 1 sphere.

I still don't feel confident about my grasp of whats happening during the computation of the mandelbulb, in particular this whole idea of turning a smooth iteration function into a distance estimate.

The DE formula usually used gives an

estimate. It looks to me like the newton root finding method.

You can take a look at this:

http://www.evl.uic.edu/hypercomplex/. It's about quaternion julia set BTW.

There is a PDF book with the distance estimate formulas and their proofs. If I've understood well, there exist a "perfect" DE for Mandebrot and

connected julia sets (with complex and quaternion) but wich is practically difficult to use. The DE formula usually used (G/G') is an approximation. The DEs for the mandelbulb (and its relatives

) have, to my knowledge, never been proved.

Nevertheless, I want to wager a guess as to what the "right" correction for pole artifacts would be.

Picture a globe with the standard spherical coordinate grid of latitude and longitude. At the equator, the grid cells are nearly perfect squares. Towards the poles, the grid cells turn rectangular, getting taller and taller. Until eventually, at the pole, the final row of grid cells is triangular, with the pole as the common tip of all triangles.

In the case of the 2d Mandelbrot, where the equations for distance estimation were originally derived, the situation is like an equator everywhere. No poles, no distortions anywhere. This is the assumption that the distance estimator is based on.

I believe the "correct correction" for the Mandelbulb distance estimates would be to shrink them proportionally to the shorter edge of a spherical grid cell at the respective latitude, to compensate for the rectangular distortion. So you would use a conical orbit trap, keeping track of the closest position to a pole. Then compute the latitude of that closest position, and shrink the distance estimate by cos(latitude). With a bit of luck, this compensation of all orbits (not just those that move close to a pole) should remove the need for global downscaling of distance estimates.

This makes sens but using cos(latitude) gives me too small estimates. I think that the contribution of each trap should be weighted with a function of the magnitude. Will see what it gives.