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首页> 外文期刊>Journal of Zhejiang University. Science, A >Generalization of 3D Mandelbrot and Julia sets
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Generalization of 3D Mandelbrot and Julia sets

机译：3D Mandelbrot和Julia Sets的概括

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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.
机译：为了进一步丰富3D Mandelbrot和Julia集合的形式，本文首先呈现了通过利用标准四元数代数的离散修改来产生三维分形集的方法，并分析它们的限制。为了克服这些限制，提出了一种基于名为Ternary代数的3D数字系统生成3D分形集的新方法。理论分析和实验结果都表明，基于三元的方法的方法优于任何一种基于Quad-代数的方法，包括本文中提出的前两种方法，因为它更直观，耗时耗时且耗时较少完全控制所得集合的几何结构。基于周期检查的射线铸造算法是开发的，该算法通过获取高质量分形图像并用于呈现在我们实验中产生的所有分形集。希望本文进行的调查将导致3D Mandelbrot和Julia概括的新观点以及其他确定性3D分形的产生。

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