3D rendering of fractal images from complex functions using potential functions

Abstract

Computer graphics plays an important role in rapid development and acceptance of fractal images visualizing the Mandelbrot set and Julia sets from a complex function. In particular, computer rendering of fractal images becomes a central tool to obtain nice fractal images and also to provide an aid for understanding the dynamical behavior of a complex function. In this thesis, we present how to visualize fractal images from complex functions. To obtain nice fractal images, we consider two families of functions which are motivated by Gujar [GB91, GBV92] and Kim [KK93, Kim92], respectively. The first one is a family of functions $f\alpha,c(z) = z^\alpha + c$ where $\alpha$ is a real number. The other function is the Newton form of an equation exp$(-\alpha\frac{zeta+z}{\zeta-z}) - 1 = 0 \mbox{where} \mid\zeta\mid = 1 and \alpha > 0$. To apply an advanced visualization technique, we first establish an efficient real valued function indicating the diverging speed of the orbit of a complex point. If the real-valued function is continuous, a 3D fractal object is constructed by giving the height to each complex point according to its corresponding real value. For visualizing a 3D fractal object, advanced rendering techniques such as ray tracing require detection of boundary points of the object and computation of the normal vectors at these points. However, since the boundary of a 3D fractal object has very complicated shapes around a Julia set and the Mandelbrot set, the points on the boundary and their normal vectors are hard to be determined. This thesis presents a method for detecting boundary points of a 3D fractal object and effectively approximating the normal vector at a boundary point. Under the iteration of a function, if the point at infinity plays a role of an attractor and a filled-in Julia set is connected, then the boundary of a fractal object for the complex function is represented by closed curves in which the orbits of all points on a curve have the sa...