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| | Q20: | How can 3-D fractals be generated? |
| | A20: | A common source for 3-D fractals is to compute Julia sets with |
| quaternions instead of complex numbers. The resulting Julia set is four |
| dimensional. By taking a slice through the 4-D Julia set (e.g. by fixing one |
| of the coordinates), a 3-D object is obtained. This object can then be |
| displayed using computer graphics techniques such as ray tracing. |
| |
| Frank Rousell's hyperindex of 3D images |
| http://www.cnam.fr/fractals/mandel3D.html |
| |
| 4D Quaternions by Tom Holroyd |
| http://bambi.ccs.fau.edu/~tomh/fractals/fractals.html |
| |
| The papers to read on this are: |
| 1. J. Hart, D. Sandin and L. Kauffman, Ray Tracing Deterministic 3-D |
| Fractals, SIGGRAPH, 1989, pp. 289-296. |
| 2. A. Norton, Generation and Display of Geometric Fractals in 3-D, |
| SIGGRAPH, 1982, pp. 61-67. |
| 3. A. Norton, Julia Sets in the Quaternions, Computers and Graphics, 13, 2 |
| (1989), pp. 267-278. |
| |
| Two papers on cubic polynomials, which can be used to generate 4-D |
| fractals: |
| 1. B. Branner and J. Hubbard, The iteration of cubic polynomials, part I., |
| Acta Math 66 (1988), pp. 143-206. |
| 2. J. Milnor, Remarks on iterated cubic maps, This paper is available from |
| ftp://math.sunysb.edu/preprints/ims90-6.ps.Z. Published in 1991 SIGGRAPH |
| Course Notes #14: Fractal Modeling in 3D Computer Graphics and Imaging. |
| Instead of quaternions, you can of course use hypercomplex number such as |
| in "FractInt", or other functions. For instance, you could use a map with more |
| than one parameter, which would generate a higher-dimensional fractal. |
| Another way of generating 3-D fractals is to use 3-D iterated function |
| systems (IFS). These are analogous to 2-D IFS, except they generate points in |
| a 3-D space. |
| A third way of generating 3-D fractals is to take a 2-D fractal such as the |
| Mandelbrot set, and convert the pixel values to heights to generate a 3-D |
| "Mandelbrot mountain". This 3-D object can then be rendered with normal |
| computer graphics techniques. |
| POV-Ray 3.0, a freely available ray tracing package, has added 4-D fractal |
| support. It takes a 3-D slice of a 4-D Julia set based on an arbitrary 3-D |
| "plane" done at any angle. For more information see the POV Ray web site at |
| http://www.povray.org/ . |
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