Fractal and Surface Modeling

Digital surface representation from a set of three-dimensional samples is an important issue of computer graphics that has applications in different areas of study such as engineering, geology, geography, meteorology, medicine, etc. The digital model allows important information to be stored and analyzed without the necessity of working directly with the real surface. In addition, we can integrate products from digital terrain model (DTM) and other data in a geospatial information system (GIS) environment. The objective of this work is to model surfaces from a set of scattered three dimensional samples. The basic structure used to represent the surface is the triangulated irregular network (TIN). Another goal of the paper is evaluation of the quality of digital terrain models for representing spatial variation. This work presents stochastic methods for triangular surface fitting. One of the most popular stochastic models to represent curves and surfaces are based on fractal concept. A fractal is a geometrical or physical structure having an irregular or fragmented shape at all scales of measurement. In addition, a fractal is based on self-similarity concept indicating that each part of its structure is similar to the whole. Brownian motion is the most popular model used to perform fractal interpolations from a set of samples. The Fractional Brownian motion (FBM), derived from Brownian motion, can be used to simulate topographic surfaces. FBM provides a method of generating irregular, self-similar surfaces that resemble topography and that have a known fractional dimension. Fractal concept has been used for optimum sampling in generating a digital terrain model. Results of the research have shown that the method can be successfully used in DTM generation. In addition fractals allow us to create realistic surfaces in shorter time than with exact calculations. Another advantage of the fractal concept is the possibility of computing surfaces to arbitrary levels of detail without increasing size of the database.