page_title: Plotting algorithms for the Mandelbrot set
section_title: Interior distance estimation
hierarchical_section_title: Plotting algorithms for the Mandelbrot set / Advanced plotting algorithms / Distance estimates / Interior distance estimation
caption_attribution_description: English: Original description: Just a printscreen from my app to illustrate the math formulas.
context_page_description: There are many programs and algorithms used to plot the Mandelbrot set and other fractals, some of which are described in fractal-generating software. These programs use a variety of algorithms to determine the color of individual pixels efficiently.
context_section_description: It is also possible to estimate the distance of a limitly periodic (i.e., inner) point to the boundary of the Mandelbrot set. The estimate is given by
  where
   is the period,
   is the point to be estimated,
   is the complex quadratic polynomial 
   is the -fold iteration of , starting with 
   is any of the  points that make the attractor of the iterations of  starting with ;  satisfies ,
  , ,  and  are various derivatives of , evaluated at .
  Analogous to the exterior case, once b is found, we know that all points within the distance of b/4 from c are inside the Mandelbrot set.
  There are two practical problems with the interior distance estimate: first, we need to find  precisely, and second, we need to find  precisely.
  The problem with  is that the convergence to  by iterating  requires, theoretically, an infinite number of operations.
  The problem with any given  is that, sometimes, due to rounding errors, a period is falsely identified to be an integer multiple of the real period (e.g., a period of 86 is detected, while the real period is only 43=86/2). In such case, the distance is overestimated, i.e., the reported radius could contain points outside the Mandelbrot set.
caption: Pixels colored according to the estimated interior distance
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