Coloring Techniques

The master’s thesis entitled On Fractal Coloring Techniques studies the mathematics of selected algorithms used to calculate colors in fractal images.

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The algorithms as well as the image discussed in the appendix are available in the Ultra Fractal Formula Database – see Stripes in jh.ucl and Pro Gradu Illustration in jh.upr.

The reader is assumed to be familiar with calculus and complex analysis. Calculation of fractal images is explained only in the extent necessary to introduce the main concepts and notation.

Content Highlights

  • A mathematical description of how the smooth iteration count works
  • The secrets of Triangle Inequality Average and Curvature Average revealed
  • An appendix illustrating the use of the discussed colorings on a fractal image


Härkönen, Jussi: On Smooth Fractal Coloring Techniques

Master’s Thesis, Department of Mathematics, Åbo Akademi University, Turku,
2007, 61 pages.

The thesis was carried out under the supervision of Professor Göran Högnäs.

Keywords: coloring function, coloring algorithm, escape time algorithm, fractal, fractal art, ultra fractal, truncated orbit, iteration count, triangle inequality average, curvature average, stripe average.

This work studies the mathematics of selected techniques for coloring fractal images. The classic escape time algorithm is extended by adding the concepts of coloring, palette and index functions. The coloring function is evaluated for each pixel of an image, whereas palette and index functions map this value to the final RGB color. In addition to good performance, also smoothness and the possibility to adjust the visual appearance are desirable characteristics of coloring functions.

The Smooth Iteration Count coloring is the smooth equivalent of the classic Iteration Count coloring. Its continuity and smoothness are studied thoroughly. The Smooth Iteration Count coloring exhibits small discontinuities at the iteration boundaries of a fractal, and a method for calculating an upper bound for the magnitude of the discontinuity is presented.

Average colorings are a family of coloring functions that use the decimal part of the smooth iteration count to interpolate between average sums. In addition to linear interpolation, a smooth Catmull-Rom spline interpolation method can be used.

The Triangle Inequality Average and Curvature Average colorings are presented and analyzed as examples of branching average colorings. Both colorings exhibit a similar tree-like branching structure, and the three mathematical properties that lead to this kind of structure are presented. A new coloring called the Stripe Average is introduced based on the behavior of the Triangle Inequality Average coloring.