Design of Random and Deterministic Fractal Surfaces from Voronoi Cells

Abstract

We show a fractal surface generation method that, unlike other methods, generates both random and deterministic fractals that model natural and architectural elements. The method starts with a succession of sets of sites, which determine, by means of a metric, a succession of Voronoi tessellations of the region where the fractal is defined. For each element of the tessellation sequence we define a tessellation function which depends on each tile. This generates a succession of tessellation functions that will be the parameter of the same seed function. Finally, the fractal is generated by a weighted sum of the seed function evaluated on each value of the succession of parameters. If the sites used to generate the Voronoi tessellation are random, natural elements such as mountains, craters, lakes, etc. are generated; if they are deterministic, architectural and decorative elements are generated. In addition, the designers can control the morphology of the generated fractal by simply varying the metric.