Distance Estimator algorithm (500x500)


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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(zo) = limk→∞ log|zk|/nk
    |G'(zo)| = lim k→∞ |dzk/dzo| / (nk|zk|)
.
For quadratic maps (n = 2)
    dzk/dzo = 2kzk-1 ... zo .
When one of zi is small (e.g. in the center of the picture at zo = 0) the distance estimate diverges therefore we get the central spot and its preimages.

The applet use 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 15 November 2002