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8.3: Distance Estimator algorithms

  • Page ID
    102626

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    8.3.1.png
    Figure \(\PageIndex{1}\)

    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

    8.3.2.png
    Figure \(\PageIndex{2}\)
    8.3.3.png
    Figure \(\PageIndex{3}\)

    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.

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

     


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