Fractals in Computer Graphics

1. Fractals in Computer Graphics

2. Fractals are special type of mathematical functions that give rise to stunning images The term “fractal” was coined by Mandelbrot in 1975 to describe self-similar functions but the underlying mathematics is much older In this section we will give a brief overview of the mathematics behind fractals and focus on the creation of fractal images Introduction

3. Early History • The idea of “space filling curves” was explored by Peano and Hilbert in the 1890s • The goal was to define a a continuous mapping from the unit interval onto the unit square • This space filling curve goes through every point in the unit square but does not cross itself

4. Early History • A “Peano curve” (1890) is a curve that passes through every point of the unit square

5. Early History • A “Hilbert curve” (1891) is another example of a continuous space filling curve

6. Early History • Hilbert curves are defined in terms of the following recursive definitions: An = Bn-1 | An-1| An-1 | Cn-1 Bn= An-1 | Bn-1| Bn-1 | Dn-1 Cn= Dn-1 | Cn-1| Cn-1 | An-1 Dn= Cn-1 | Dn-1| Dn-1 | Bn-1 with A0 B0 C0 D0

7. In 1967 Lewis Fry Richardson was studying the length of country borders and found that the measuring stick effects answer Richardson 11.5*200 = 2300km 28*100 = 2800km 70*50 = 3500km

8. The term “fractal” was coined in 1975 by Mandelbrot to describe self-similar functions Mandelbrot sets The first picture of the Mandelbrot set, by Robert W. Brooks and Peter Matelski in 1978

9. During the 1980s Mandelbrot built expanded the mathematical foundation and applications of fractals Mandelbrot sets The cover article of Scientific American in August 1985 brought fractals to wider range of people (including me)

10. The fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a fractal pattern changes with the scale at which it is measured. The dimension of ‘non-fractal objects’ is equal to the dimension of the space it is drawn in. Fractal objects will have fractional dimensions Fractal Dimension

11. D= log(N) / log(S) where N =number of new edges S = number of subdivisions Fractal Dimension

12. The Koch snowflake can be constructed by starting with an equilateral triangle, then recursively altering each line segment as follows: divide the line segment into three segments of equal length. draw an equilateral triangle that has the middle segment from step 1 as its base and points outward. remove the line segment that is the base of the triangle from step 2. Koch snowflake

13. The length of the curve for a Koch snowflake increases by a factor of (4/3) in each iteration since we replace each line segment by 4 segments 1/3 the length of original The fractal dimension of this infinite length curve is log(4)/log(3)= 1.26 Koch snowflake

14. Fractals are used to create many beautiful images and also to create realistic looking models of mountains, trees, clouds and other objects with ‘random looking’ shapes Conclusions