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Hi

I wrote a new renderer that shows the process during rendering.. and I found that the analytical DE sometimes do not converge at all it simply oscillates. I think this is at the core of the problem of overestimating the distance.

The oscillation seem to be then DE overshoots to negative values it then steps back to the positive side and overshoots to negative again. This is no matter how many iterations you do in the DE estimate, it helps adding more iterations though.

I added a detection that checks if the previous step was negative and the current one positive..thus it must have overshoot and come back. If this is the case I added a reduction factor to the step with the factor initially set to 1.0. Every time oscillation is detected (for the same ray) I set reduction*=0.998

I think this causes convergence to the right spot since even though DE overshoots its sign correctly say if I you are  infront of or behind the mandelbulb surface..

See this video of the mandelbrot generated with special case of the mandelbulb equation and the ordinary DE.
The fluctuating area that lasts a long time is oscillating DE (I could make it converge faster..but that would ruin the illustration!)

<a href="http://www.youtube.com/v/z5eQbF6lEuI&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/z5eQbF6lEuI&rel=1&fs=1&hd=1</a>


DE can bcome negative then |Z| is less than 1.0, for instance log(0.5) = -0.30..
dist= 0.5*|Z|*log(|Z|)/|DZ|

Your results are correct and symptomatic of the misuse of DE.

The DE method you are using is *only* valid "outside" so if you hit maximum iterations then you should step backwards to find the exact boundary and *not* try and use the DE value from that position.

Note that there are some posts on the "Mandeliers", these are artefacts produced by incorrectly using the outide DE algorithm "inside".

This DE value is only accurate for the infinite attractor i.e. only "outside" so although you can get negative values from "inside" they will not be the correct magnitude for stepping and you can also get positive values for "inside" points which effectively say that you haven't reached the solid threshold when in fact you've already passed it.

I realized that above method counts on the sign of the distance to be right relative to the starting point of each estimation..

Thus that it tries to go in the right direction... as long as this holds the method will converge to the right surface..

Even though distance can be both pos and neg inside and has the wrong magnitude.

Remains to be proven though  

I should add that the "delta DE" methods are directional unlike the analytical method.