File:Mandel bd.jpg
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This is a file from the Wikimedia Commons |
Summary
| Description |
English: binary decomposition of level sets of exterior of Mandelbrot set = BDM/M "Our images look noisy and grainy near the boundary of the Mandelbrot set. The escape time bands get closer and closer, while the pixel spacing is fixed. The pixel grid samples isolated points of a mathematically abstract image defined on the continuous plane. The Nyquist-Shannon sampling theorem shows that sampling isolated points from a continuum is a valid approximation only so long as the values don’t change too quickly between the points. Aliasing occurs when the values do change too quickly compared to the sampling rate, with the grainy noisy visual effects as we have seen. Because the escape time bands increase in number without bound as we approach the boundary of the Mandelbrot set, no sampling rate can be high enough." Claude Heiland-Allen |
| Date | |
| Source | Own work |
| Author | Adam majewski |
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C source code
It is a console C program ( one file) It can be compiled under :
- windows ( gcc thru Dev-C++ )
- linux and mac using gcc :
gcc main.c -lm
it creates a.out file. Then run it :
./a.out
It creates ppm file in program directory. Use file viewer to see it.
/*
c program:
1. draws Mandelbrot set for Fc(z)=z*z +c
using Madelbrot algorithm ( boolean escape time )
-------------------------------
2. technic of creating ppm file is based on the code of Claudio Rocchini
http://en.wikipedia.org/wiki/Image:Color_complex_plot.jpg
create 24 bit color graphic file , portable pixmap file = PPM
see http://en.wikipedia.org/wiki/Portable_pixmap
to see the file use external application ( graphic viewer)
---------------------------------
*/
#include <stdio.h>
#include <math.h>
int main()
{
/* screen ( integer) coordinate */
int iX,iY;
const int iXmax = 2000;
const int iYmax = 2000;
/* world ( double) coordinate = parameter plane*/
double Cx,Cy;
const double CxMin=-2.5;
const double CxMax=1.5;
const double CyMin=-2.0;
const double CyMax=2.0;
/* */
double PixelWidth=(CxMax-CxMin)/iXmax;
double PixelHeight=(CyMax-CyMin)/iYmax;
/* color component ( R or G or B) is coded from 0 to 255 */
/* it is 24 bit color RGB file */
const int MaxColorComponentValue=255;
FILE * fp;
char *filename="new1.ppm";
char *comment="# ";/* comment should start with # */
static unsigned char color[3];
/* Z=Zx+Zy*i ; Z0 = 0 */
double Zx, Zy;
double Zx2, Zy2; /* Zx2=Zx*Zx; Zy2=Zy*Zy */
/* */
int Iteration;
const int IterationMax=2000;
/* bail-out value , radius of circle ; */
const double EscapeRadius=1000;
double ER2=EscapeRadius*EscapeRadius;
/*create new file,give it a name and open it in binary mode */
fp= fopen(filename,"wb"); /* b - binary mode */
/*write ASCII header to the file*/
fprintf(fp,"P6\n %s\n %d\n %d\n %d\n",comment,iXmax,iYmax,MaxColorComponentValue);
/* compute and write image data bytes to the file*/
for(iY=0;iY<iYmax;iY++)
{
Cy=CyMin + iY*PixelHeight;
if (fabs(Cy)< PixelHeight/2) Cy=0.0; /* Main antenna */
for(iX=0;iX<iXmax;iX++)
{
Cx=CxMin + iX*PixelWidth;
/* initial value of orbit = critical point Z= 0 */
Zx=0.0;
Zy=0.0;
Zx2=Zx*Zx;
Zy2=Zy*Zy;
/* */
for (Iteration=0;Iteration<IterationMax && ((Zx2+Zy2)<ER2);Iteration++)
{
Zy=2*Zx*Zy + Cy;
Zx=Zx2-Zy2 +Cx;
Zx2=Zx*Zx;
Zy2=Zy*Zy;
};
/* compute pixel color (24 bit = 3 bajts) */
if (Iteration==IterationMax)
{ /* interior of Mandelbrot set = black */
color[0]=0;
color[1]=0;
color[2]=0;
}
/* exterior of Mandelbrot set = binary decomposition of level sets of escape time */
else if (Zy>0)
{
color[0]=0; /* Red*/
color[1]=0; /* Green */
color[2]=0;/* Blue */
}
else
{
color[0]=255; /* Red*/
color[1]=255; /* Green */
color[2]=255;/* Blue */
};
/*write color to the file*/
fwrite(color,1,3,fp);
}
}
fclose(fp);
return 0;
}
GLSL code
// fragmentarium
#include "2D.frag"
#group Simple Mandelbrot
// maximal number of iterations
uniform int iMax; slider[1,100,1000] // increase iMax
// er2= er^2 wher er= escape radius = bailout
uniform float er2; slider[4.0,1000,10000] // increase er2
// compute color of pixel
vec3 color(vec2 c) {
vec2 z = vec2(0.0); // initial value
// iteration
for (int i = 0; i < iMax; i++) {
z = vec2(z.x*z.x-z.y*z.y,2*z.x*z.y) + c; // z= z^2+c
if (dot(z,z)> er2) // escape test
// exterior
if (z.x>0){ return vec3( 1.0);} // upper part of the target set
else return vec3(0.0); //lower part of the target set
}
return vec3(0.0); //interior
}
Licensing
|
I, the copyright holder of this work, release this work into the public domain. This applies worldwide. In some countries this may not be legally possible; if so: I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law. |
File history
Click on a date/time to view the file as it appeared at that time.
| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 15:03, 22 December 2007 | | 2,000 × 2,000 (824 KB) | Soul windsurfer | {{Information |Description=binary decomposition of level sets of exterior of Mandelbrot set |Source=self-made |Date=2007.12.21 |Author= Adam majewski |Permission= |other_versions= }} |
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