Distance Estimator algorithm

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Distance Estimator algorithm is used in this applet. Color is "proportional" to log(dc), where dc is the approximate distance between the point c and the nearest point in the Mandelbrot set.

"DE" algorithm and Julia sets

The distance estimate for Julia sets is very close [1] to the ratio |G|/|G'|, where
G(z_o) = lim_k→∞ log|z_k|/n^k
|G'(z_o)| = lim _k→∞ |dz_k/dz_o| / (n^k|z_k|) .
For quadratic maps (n = 2)
dz_k/dz_o = 2^kz_k-1 ... z_o .
When one of z_i is small (e.g. in the center of the picture at z_o = 0) the distance estimate diverges therefore we get the central spot and its preimages.

This script uses the Boundary Tracing algorithm too. Therefore these spots disappear under enlargement.

[1] John W. Milnor "Dynamics in One Complex Variable" Appendix G. Remarks on Computer Graphics.

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updated 12 Sep 2013