Drawing the Mandelbrot Set Sep. 19 Dae-Eun Hyun 3D MAP Lab.

What is the Mandelbrot Set? Def. The Mandelbrot Set M is the set of all complex numbers c that produces a finite orbit of 0.

What is the Mandelbrot Set? The points of the complex plane in two categories Points inside the Mandelbrot Set Points outside the Mandelbrot Set

Computing the Mandelbrot Set How we can decide the convergency? Set the maximum magnitude of | Zk | Typically 2 How many we have to iterate? Set some upper limit Number on the maximum number of iterations Typically 100 ~ 400

Computing the Mandelbrot Set C inside the set C outside the set

Drawing the Mandelbrot Set Display M on the raster graphics Set up a correpondence between each pixel on the display and a value of C, and the iteration number for that C-value is found. Example Bright yellow to C near outside the set Dimmer yellow to C farther away from the set Deep Blue to C have the small iteration Num

Drawing the Mandelbrot Set Color Intensity Red Blue Num Float v = d/ float Num; glColor3f(v*v, v*v, v, 0.2);

Drawing the Mandelbrot Set How to associate a pixel with a specific complex value of C Image : the Number of Rows the Number of Columns

Drawing the Mandelbrot Set Pseudocode for Drawing for( j = 0; j < rows; j++) for( i = 0; i < cols; I++) { find the correspondence c-value for Pixel (i,j); estimate the iteration Number of the orbit; find Color determined the iteration Number; setPixel(j, i, Color); }