Iterated Fractal Based on Distance Ratio

DOI：

 作者 单位 张锡哲 吉林大学 计算机科学与技术学院，吉林 长春 130012 符号计算与知识工程教育部重点实验室（吉林大学），吉林 长春 130012 吕天阳 吉林大学 计算机科学与技术学院，吉林 长春 130012 符号计算与知识工程教育部重点实验室（吉林大学），吉林 长春 130012 哈尔滨工程大学 计算机科学与技术系，黑龙江 哈尔滨 150001 王钲旋 吉林大学 计算机科学与技术学院，吉林 长春 130012 符号计算与知识工程教育部重点实验室（吉林大学），吉林 长春 130012

由于逃逸时间算法不能绘制函数收敛区域,所以现有的分形图大都存在大片的黑色区域.提出一种新的构造分形图的方法:距离比值迭代法.该方法采用两点迭代,利用其距离比值的收敛次数来绘制分形图.利用距离比值迭代法绘制了复映射zzα+c的广义M-J集并分析其构图性质.距离比值广义M-J集的内部收敛区域具有复杂的细节和自相似结构,当α>0时其外部边界与经典M-J集一致,当α<0时能够绘制出经典M-J集所没有的复杂结构.

The escape time algorithm cannot render the convergence region of mapping, so there are some black regions in escape time fractal. In this paper, a novel method is presented to construct fractal image, which is named the distance ratio iteration method. This method performs iteration on two points and render fractal image by using their distance ratio convergence times. Taking complex mapping zzα+c as example, the generalized Mandelbrot and Julia sets are constructed based on distance ratio and their visual properties are analyzed. The result fractal image has complex and self-similarity structure in inner convergence region. It is proved that the boundary of distance ratio fractal is the same as M-J set when α>0, and some visual structure of it with various exponent α are discussed. When α<0, the generalized Mandelbrot and Julia set based on distance ratio have some complex structures which M-J set does not have.
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