论Mandelbulb的代数基础
On the Algebraic Foundation of the Mandelbulb
AI总结：
该研究基于纯四元数定义球积，构建相关代数结构，推广Mandelbrot集，得到与Mandelbulb视觉一致的有界集及bulbic Mandelbrot集，明确两类集合的生成方式与动力学特性。
摘要：
本文利用四元数与球坐标对Mandelbrot集（曼德博集合）进行推广。具体而言，我们使用纯四元数定义了一种球积，该乘积受复数乘法启发，将球坐标的角度相加、半径相乘。我们证明，带有球积的纯四元数的代数结构是一个交换含幺广群。随后，我们提出了Mandelbrot集的若干推广形式，其中包括一个视觉上与所谓Mandelbulb（曼德尔球）完全一致的集合，我们证明该集合是有界的，且可通过逃逸时间算法生成。我们还定义了另一种推广形式——bulbic Mandelbrot集（球茎状曼德博集合），并证明其一个二维截面具有与Mandelbrot集相同的动力学特性，且该集合仅需使用四元数乘积即可生成，无需用到球积。
英文摘要：
In this paper, we generalize the Mandelbrot set using quaternions and spherical coordinates. In particular, we use pure quaternions to define a spherical product. This product, which is inspired by the product of complex numbers, add the angles and multiply the radii of the spherical coordinates. We show that the algebraic structure of pure quaternions with the spherical product is a commutative unital magma. Then, we present several generalizations of the Mandelbrot set. Among them, we present a set that is visually identical to the so-called Mandelbulb. We show that this set is bounded and that it can be generated by an escape time algorithm. We also define another generalization, the bulbic Mandelbrot set. We show that one of its 2D cuts has the same dynamics as the Mandelbrot set and that we can generate this set only with a quaternionic product, without using the spherical product.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
