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Journal of Zhejiang University SCIENCE A 2007 Vol.8 No.1 P.134~141

10.1631/jzus.2007.A0134


Generalization of 3D Mandelbrot and Julia sets


Author(s):  CHENG Jin, TAN Jian-rong

Affiliation(s):  State Key Laboratory of CAD & CG, Zhejiang University, Hangzhou 310027, China

Corresponding email(s):   cjinpjun@zju.edu.cn

Key Words:  Mandelbrot set, Julia set, Fractal, Ray-casting, Quad algebra, Ternary algebra


CHENG Jin, TAN Jian-rong. Generalization of 3D Mandelbrot and Julia sets[J]. Journal of Zhejiang University Science A, 2007, 8(1): 134~141.

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author="CHENG Jin, TAN Jian-rong",
journal="Journal of Zhejiang University Science A",
volume="8",
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pages="134~141",
year="2007",
publisher="Zhejiang University Press & Springer",
doi="10.1631/jzus.2007.A0134"
}

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%A CHENG Jin
%A TAN Jian-rong
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%DOI 10.1631/jzus.2007.A0134

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T1 - Generalization of 3D Mandelbrot and Julia sets
A1 - CHENG Jin
A1 - TAN Jian-rong
J0 - Journal of Zhejiang University Science A
VL - 8
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SP - 134
EP - 141
%@ 1673-565X
Y1 - 2007
PB - Zhejiang University Press & Springer
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DOI - 10.1631/jzus.2007.A0134


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.

Darkslateblue:Affiliate; Royal Blue:Author; Turquoise:Article

Reference

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[12] Norton, A., 1989. Julia sets in the quaternions. Computers and Graphics, 13(2):267-278.

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Open peer comments: Debate/Discuss/Question/Opinion

<1>

hoshang<shahinkey@yahoo.com>

2010-09-27 10:00:58

I am phd student and i work on shadow and ray casting. If you can help me please send me some information about ray casting,ray tracing and shadow.
thank you in advance

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