FRACTAL AND SURFACE MODELING 
M. Rahnemoonfar '. M.R. Delavar , L. Hashemi 
Department of surveying and Geomatics Engineering, Engineering Faculty, University of Tehran, Tehran, Iran 
rahnemon(@ut.ac.ir 
mdelavar@ut.ac.ir 
  
Commission IV, WG IV/6 
KEY WORDS: Land, Modelling, Sampling, DEM/DTM, Three-dimensional 
ABSTRACT: 
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. 
1. INTRODUCTION 
So far we have assumed that sharp boundaries or smooth shapes To better understand this concept we describe steps for 
exist for real entities. This assumption reflects a map model or producing the geometrical shape of natural objects like 
geometric bias rather than an appropriate model to represent snowflake. Suppose that we have a triangle. By dividing each 
nature. side to three parts and replacing the middle part with two equal 
Smoothed curves and surfaces are subjects of Euclidean parts a polygon with twelve sides will be generated. By 
geometry and are adequate to represent artificial shapes like repeating the above stage for each side, at step two there are 
parts of mechanical and aeronautical projects, furniture, toys, forthy-eight sides. At each step, the number of sides are 
etc. Natural objects like clouds, coastlines and mountains have multiplied by four. So for an initiator perimeter of length 1, the 
irregular or fragmented features. 4 N 
These are better represented by the Fractal geometry that was perimeter becomes / 2 | which number evidently tends to 
first formalized by Mandelbrot (1982). 
M: infinity, although the area tends to a finite limit. The self- 
Fractal geometry has enough capability to represent more 7 
adequately than Euclidean geometry real world entities that are similarity ratio isl à 
not smoothly formed, as in the case with most natural objects. 
The word fractal implies properties as in fraction or fragmented. Therefore, fractal geometry has promise for some of the 
In essence fractal geometry has ideas of fragmentation and self- requirements of spatial information systems. Two-dimensional 
similarity (Laurini and Thompson 2002). stochastic interpolations are useful for terrain modeling 
Even though objects may be rough or irregular, there is (Felgueiras and Goodchild 1995; Goodchild and Mark 1987). 
fragmented, they may at the same time have some similar One dimensional application use of fractal concept is for 
semblance of shape or pattern when viewed from different coastlines (Cheng et al 2001) or boundaries of entire continents. 
distance. Self-similarity is symmetry across different scales; Fractals may also be used for image error analysis (Kolibal and 
there are patterns within patterns. Or, as Mandelbort says, Monde 1998), assigning color palette (Cheng and Qingmou 
fractals are geometric shapes that are equally complex in their 
details as in their overall form (Mandelbrot 1982). 
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