Continuous Dwell
Robert P. Munafo, 2023 Mar 20.
"Continuous" dwell refers to any method of calculating dwell that gives a continuous value, i.e. a value that varies smoothly rather than jumping between integer values. This allows for more variety in coloring algorithms, and is essential for making a 3D shape of the Mandelbrot set, or for visualisation of such things as ridgelines.
The simplest method is to choose a fairly large escape radius (such as 1000), then perform the ordinary iteration algorithm. Let Z_{n} be the value of the iterate that exceeds the escape radius. The normal dwell value would be n (as shown in the pseudocode in the escapeiterations article). The "continuous dwell" value is given by
D = n + log_{2}(log_{2}(Z_{n}))  log_{2}(log_{2}(EscapeRadius))
Variation
The formula above will give something that is devoid of any dwell band boundaries, because the part with the two log_{2}(log_{2}()) expressions will be a fractional value from 0.0 to 1.0, and each band will blend smoothly into the next. You may wish to create distinct color transitions at the integer dwell bands by multiplying the log_{2}(log_{2}()) section by a constant K so that it goes from 0.0 to K.
Comparison


revisions: 20221202 oldest on record; 20230320 link to 3D and ridgelines, and Wikipedia "continuous function"
From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 19872023.
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This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2023 Mar 21. s.27