基于三维多项式映射的广义Julia集表示与绘制

研究了基于三维多项式映射的三维广义Julia集表示方法.从理论上分析并证明了三维多项式映射满足等变的条件,精确地给出了关于正四面体群和正八面体群具有旋转不变对称性的两类三维等变映射的具体公式,在此基础上讨论并证明了三维多项式映射的广义Julia集所具有的性质.提出了基于逃选距离色彩调配的光线跟踪体绘制算法,对给定三维空间中属于Julia集的离散点根据其逃逸距离赋予颜色和不透明度,并采用光线跟踪法进行体绘制.实验结果表明,利用三维多项式映射来构造三维Julia集,不仅可以根据映射的性质预知Julia集的总体结构特征,并且能够通过调控映射的参数来获得多种具有不同旋转对称结构的Julia集,因而有效地克服了现有三维分形集生成方法所构造的分形集包含信息量少、形状结构单一和分形形状无法预测等缺陷.进一步地,三维多项式映射可以应用于其他三维分形的构造,从而为三维分形的生成提供一个新的有效途径.

Representation and Rendering of General Julia Sets Based on Tri-Dimensional Polynomial Maps

A new approach for the representation of 3D general Julia sets is put forward on the basis of tri-dimensional polynomial maps. The condition for a 3D polynomial map to be equivariant is theoretically analyzed and proved. The equations of two classes of 3D polynomial maps that are equivariant with respect to the rotational symmetries of either a regular tetrahedron group or a regular octahedron group are strictly given, on the basis of which the properties of the general Julia sets created by these 3D polynomial maps are discussed and proved. A ray-tracing volume rendering algorithm, which defines the color and opacity of every discrete point within a Julia set according to its escaping distance, is proposed in order to acquire high quality 3D fractal images. Experimental results demonstrate that the approach of generating 3D Julia sets from 3D polynomial maps not only enables us to predict the characteristics of Julia sets according to the properties of the maps, but also makes it possible for us to obtain various kinds of Julia sets with different rotational symmetries by altering the parameters of the maps. Consequently, drawbacks such as monotone structure of the resulting fractals and inability to predict fractal shape in the existing methods for generating 3D fractal sets can be effectually avoided. Furthermore, the method of generating 3D Julia sets by 3D polynomial maps can be applied to the construction of other 3D fractals, and hence would result in a different perspective for the generation of 3D fractals.