基于逃逸时间算法的三维分形图绘制方法

Rendering 3D Fractals Based on the Escape-Time Algorithm

使用经典的逃逸时间算法通常得到的是黑白的或缺乏颜色过渡的Mandelbrot集或Julia集。通过定义色彩控制球,给出一种绘制三维分形图的方法。对于确定区域内的迭代初始点,首先将动力系统的迭代逃逸点或非逃逸点映射到色彩控制球内,然后计算映射点到球心的距离;以该距离为参数,经由色彩控制函数确定迭代初始点的色彩;再以该色彩绘制迭代初始点域中的嵌入体表面,从而得到具有伪三维效果的分形图。

The Mandelbrot-set and Julia-set are two classical 2D fractals generated by the Escape-Time Algorithm （ETA） with simple color patterns, In this paper, we present a novel method to render colorful 3D fractals based on an improved ETA and Color-Control Sphere （CCS）, In the method, each point in parameter domain, to be called parameter points, is judged if it is an escaped point or not firstly, under the governing of discrete dynamic system, The point is then mapped to a mapped-point in CCS and the distance between the mapped-point and center point of CCS is evaluated, Subsequently, a color for rendering the parameter point is can be obtained by employing a predefined color-control function system with respect to the distance. All of the surface points, which are taken as a set of initial points of a dynamic system, of arbitrary shape embedded in the parameter domain are rendered with their corresponding colors, As a result, a pseudo-3D fractal can be archived on the surface of the embedded shape.