Normal Vector Based Subdivision Scheme to Generate Fractal Curves

In this paper, we firstly devise a new and general p-ary subdivision scheme based on normal vectors with multi-parameters to generate fractals. Rich and colorful fractals including some known fractals and a lot of unknown ones can be generated directly and conveniently by using it uniformly. The method is easy to use and effective in generating fractals since the values of the parameters and the directions of normal vectors can be designed freely to control the shape of generated fractals. Secondly, we illustrate the technique with some design results of fractal generation and the corresponding fractal examples from the point of view of visualization, including the classical Lévy curves, Dragon curves, Sierpiński gasket, Koch curve, Koch-type curves and other fractals. Finally, some fractal properties of the limit of the presented subdivision scheme, including existence, self-similarity, non-rectifiability, and continuity but nowhere differentiability are described from the point of view of theoretical analysis.