A Generalized Mandelbrot Set Based On Distance Ratio

Abstract

The iteration of complex function can generate beautiful fractal images. This paper presents a novel method based on the iteration of the distance ratio with two points, which generates a generalized Mandelbrot set according to distance ratio convergence times. This paper states the definition of distance ratio and its iteration. Then taking the complex function f(z)=zα+c for example, it discusses the visual structure of generalized Mandelbrot with various exponent and comparing it with Mandelbrot set generated by escape time algorithm. When exponent α>1, the outer border of DRM is same as Mandelbrot set, but has complex inner structure; when α<0, the inner border of DRM is same as Mandelbrot set, DRM is the “outer” region and complement set of Mandelbrot set, the two sets cover the whole complex plane.