Chaos and GraphicsOrbit trap rendering method for generating artistic images with cyclic or dihedral symmetry
Introduction
Pickover [1] graphically explored the Mandelbrot set with the “epsilon cross” method in which an iteration loop exits when a point in the orbit falls within the distance ε of either the real or imaginary axis, and then the pixel color is determined by the iteration counts. Carlson [2], [3] extended this idea and used several “orbit trap” rendering methods to create 3D effects in fractal images. Ye [4] used the Mandelbrot sets and Julia sets as the orbit traps and obtained some artistic fractal images. Lu et al. [6] generated artistic images with wallpaper symmetries by using the orbit trap method.
This paper extends concepts presented previously by the authors [6] to obtain artistic images with cyclic (Zn) or dihedral (Dn) symmetry by setting some symmetric orbit traps and applying the orbit-trap rendering method to the chaotic functions described in [5]. These kinds of chaotic functions are studied and constructed by Field and Golubitsky [5] to generate chaotic attractors with cyclic (Zn) or dihedral (Dn) group symmetry. A truncated form for those functions can be written in complex coordinates aswhere are real numbers, and corresponds to dihedral symmetry.
Section snippets
Description of colormaps
Two colormaps are used to generate images in this paper. One approach is given in Table 1 in [6], and the reader is directed to this reference for details. The other is shown in Table 1 of the current paper. Similar to [6], we create two new colormaps based on the two color tables, using the following formula:where C is a constant. For , the value of ColorRatio is computed by Eq. (2). Because , the RGB value
Description of orbit trap methods
In this section we present three orbit trap methods to create images with Zn or Dn symmetry. To do this, we take F(z) given in Eq. (1) as the generating function and set three symmetric orbit traps. The orbit traps possess Zn or Dn symmetry so that they are compatible with the generating function .
To determine the RGB value at the original point used for the iteration, we take the ColorRatio to be TrapRatio. The latter is defined as the ratio of the distance from the orbit point to the
Conclusion
Chaotic functions in complex plane can be generated with the symmetry of the Zn or Dn group. We take these functions as generating functions and use the orbit trap rendering method to generate artistic images with Zn or Dn symmetry. In this paper, we set three symmetric orbit traps compatible with these chaotic functions. In fact, any subset with Zn or Dn symmetry can be chosen as the orbit trap, which could be generalized to more complicated bounded area such as chaotic attractors with
Acknowledgements
This reaearch was supported by National Science Foundation of China #10371043 and Shanghai Priority Academic Discipline.
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