|Chinese Journal of Computers Full Text
|Representation of 3-D General Mandelbrot Sets Based on Ternary Number and Its Rendering Algorithm
|CHENG Jin TAN Jian-Rong
|£¨State Key Laboratory of CAD&CG, Zhejiang University, Hangzhou 310027£©
| Abstract &
| The Generation of Mandelbrot sets in more than two dimensions is a new and difficult problem in computer graphics. All the existing methods utilize quaternion algebra to iteratively compute M-sets¡¯ boundaries. This paper proposes a new approach to the problem and represents the 3-D general M-sets based on ternary number. Firstly, the M-sets¡¯ properties of the ternary mapping t¡ûtm+c for mªÅ2 are theoretically analyzed and proved. Secondly, a ray-casting volume rendering algorithm on the basis of period checking is put forward for the first time. In the new algorithm, the color, opacity and normal of every discrete point are defined according to its periodicity and the computation of a ray¡¯s intersecting with M-set is accelerated by making use of Newton-Raphson method. Finally, the 3-D M-sets generated by the ternary number and quaternion algebra are both rendered by our new algorithm. The experimental results show that several advantages such as intuitionistic, fast and controllable are obtained by using ternary mapping to generate 3-D M-sets in comparison with traditional methods that use quaternion algebra to generate them. Furthermore, the method of generating 3-D M-sets by ternary mapping can be applied to the construction of other 3-D Mandelbrot sets and Julia sets. Consequently this results in a different perspective for the generation of 3-D fractal sets.
keywords ternary number; general Mandelbrot set; volume rendering; period-checking algorithm; quaternion algebra