1. CS4395: Computer Graphics 1 Fractals Mohan Sridharan Based on slides created by Edward Angel

2. Modeling Geometric: – Meshes. – Hierarchical. – Curves and Surfaces (coming up soon!). Procedural: – Particle Systems. – Fractal. CS4395: Computer Graphics 2

3. Sierpinski Gasket Rule based: Repeat n times. As n →∞: – Area→0 – Perimeter →∞ Not a normal geometric object. CS4395: Computer Graphics 3

4. Coastline Problem What is the length of the coastline of England? There is no single answer: – Depends on length of ruler (units). If we experiment with maps at various scales we also notice self-similarity: each part looks like a whole! CS4395: Computer Graphics 4

5. Fractal Geometry Created by Mandelbrot: – Self similarity. – Dependence on scale. Leads to the idea of fractional dimension. Graftals: graphical fractal objects. CS4395: Computer Graphics 5

6. Koch Curve/Snowflake (Figure 11.12) CS4395: Computer Graphics 6 Recursive lengthening: In the limit, infinite length and discontinuous first derivative. Not a 2D object either!

7. Fractal Dimension Start with unit line, square, cube which we agree are 1D, 2D, 3D respectively under any reasonable dimension. Consider scaling each one by a h = 1/n, the smallest unit we can measure. Scale object by h and replicate k times. CS4395: Computer Graphics 7

8. How Many New Objects? Line: k = n Square: k = n 2 Cube: k = n 3 The whole is the sum of its parts implies: 8 CS4395: Computer Graphics = 1 d =

9. Examples Koch Curve: – Sub-division (scaling) of the original by a factor of 3. – Create 4 new objects. – Fractal dimension: d = ln 4 / ln 3 = 1.26186. Sierpinski gasket: – Sub-division (scaling) by a factor of 2. – Keep 3 of the 4 triangles created. – d = ln 3 / ln 2 = 1.58496. CS4395: Computer Graphics 9

10. Volumetric Examples CS4395: Computer Graphics 10 3D version of Sierpinski gasket: d = ln 4/ ln 2 = 2. One iteration of the sponge: d = ln 20 / ln 3 = 2.72683.

11. Midpoint subdivision CS4395: Computer Graphics 11 Randomize displacement using a Gaussian random number generator. Reduce displacement each iteration by reducing variance of generator.

12. Fractal Brownian Motion CS4395: Computer Graphics 12 variance ~ length -(2-d) Brownian motion d = 1.5

13. Fractal Mountains CS4395: Computer Graphics 13

14. Iteration in the Complex Plane CS4395: Computer Graphics 14

15. Mandelbrot Set Iterate on z k+1 =z k 2 +c with z 0 = 0 + j0 Two cases as k →∞: – |z k |→∞ – |z k | remains finite. If for a given c, |z k | remains finite, then c belongs to the Mandelbrot set. CS4395: Computer Graphics 15

16. Mandelbrot Set (Section 11.8.5) CS4395: Computer Graphics 16

17. Mandelbrot Set CS4395: Computer Graphics 17