复映射z←()^(-a)+c(a≥2)的广义M集及其对称周期检测法

General Mandelbrot Sets from the Complex Mapping z←()^(-a)+c(a≥2）and Its Symmetrical Period-Checking Algorithm

研究了指数为负实数的非解析复映射()()2+-aaczz的广义Mandelbrot集.分析和证明了a取不同值时该映射的广义M集所具有的性质,严格地给出了a为正整数时复映射周期1轨道稳定区域边界的参数方程.提出了对称周期检测法,根据各参数点的周期值对M集进行着色,并充分利用M集的对称性来减少绘制过程中计算周期时所需要的迭代运算.实验结果表明,新算法在获得高质量M分形图的同时具有较高的绘制速度.进一步地,新算法可以推广到其他M(Mandelbrot)集和J(Julia)集的绘制.

The general Mandelbrot sets from the non-analytical complex mapping ()czz+-a for 2a are studied in this paper. The M-sets?properties of different parameter are theoretically analyzed and proved. The parameter equation of the fixed point region抯 boundary when is a positive integer are strictly given. A symmetrical period-checking algorithm, which colors M-sets according to the period of each point in the complex C-plane, is put forward for the first time. The new algorithm takes full advantage of the M-sets?property during the rendering process and can greatly reduce the number of iterations in calculating the period of all pixel points in the drawing region. The experimental results show that both high quality and high drawing speed of the M-sets?fractal image can be acquired with the new algorithm. Furthermore, the new algorithm can be generalized to the drawing of other Mandelbrot sets and Julia sets.