论文标题
在Mandelbulb的代数基础上
On the Algebraic Foundation of the Mandelbulb
论文作者
论文摘要
在本文中，我们使用四元组和球形坐标概括了mandelbrot集。特别是，我们使用纯季度来定义球形产品。该产品的灵感来自复数的乘积，添加角度并乘以球形坐标的半径。我们表明，纯四季度使用球形产品的代数结构是交换性的Unital Magma。然后，我们提出了Mandelbrot集的几个概括。其中，我们提供了与所谓的Mandelbulb视觉相同的一组。我们证明该集合是有限的，并且可以通过逃生时间算法生成。我们还定义了另一个概括，即Bulbic mandelbrot集。我们表明，其2D切割之一具有与Mandelbrot集相同的动力学，并且我们只能使用Quaternionic产品来生成此集合，而无需使用球形产品。
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.
