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Generalization of 3D Mandelbrot and Julia sets

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Abstract

In order to further enrich the form of 3D Mandelbrot and Julia sets, this paper first presents two methods of generating 3D fractal sets by utilizing discrete modifications of the standard quaternion algebra and analyzes the limitations in them. To overcome these limitations, a novel method for generating 3D fractal sets based on a 3D number system named ternary algebra is proposed. Both theoretical analyses and experimental results demonstrate that the ternary-algebra-based method is superior to any one of the quad-algebra-based methods, including the first two methods presented in this paper, because it is more intuitive, less time consuming and can completely control the geometric structure of the resulting sets. A ray-casting algorithm based on period checking is developed with the goal of obtaining high-quality fractal images and is used to render all the fractal sets generated in our experiments. It is hoped that the investigations conducted in this paper would result in new perspectives for the generalization of 3D Mandelbrot and Julia sets and for the generation of other deterministic 3D fractals as well.

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Authors and Affiliations

  1. State Key Laboratory of CAD & CG, Zhejiang University, Hangzhou, 310027, China

    Cheng Jin & Tan Jian-rong

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  1. Cheng Jin
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  2. Tan Jian-rong
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Project supported by the National Basic Research Program (973) of China (Nos. 2004CB719402 and 2002CB312106), the National Natural Science Foundation of China (Nos. 60375020 and 50305033), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20020335112)

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Cheng, J., Tan, Jr. Generalization of 3D Mandelbrot and Julia sets. J. Zhejiang Univ. - Sci. A 8, 134–141 (2007). https://doi.org/10.1631/jzus.2007.A0134

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  • DOI: https://doi.org/10.1631/jzus.2007.A0134

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