Julia set
Generate and draw a Julia set.
- Cf
AWK
<lang AWK>
- syntax: GAWK -f JULIA_SET.AWK [real imaginary]
- converted from COBOL
BEGIN {
c_real = (ARGV[1] != "") ? ARGV[1] : -0.8
c_imaginary = (ARGV[2] != "") ? ARGV[2] : 0.156
printf("%s %s\n",c_real,c_imaginary)
for (v=1; v<=320; v+=10) {
for (h=1; h<=560; h+=10) {
x = (h - 280) / 200
y = (v - 160) / 100
plot_char = "#"
for (i=1; i<=50; i++) {
if (plot_char == "") {
break
}
z_real = ((x * x) - (y * y)) + c_real
z_imaginary = ((x * y) * 2) + c_imaginary
if (z_real ^ 2 > 10000) {
plot_char = ""
}
x = z_real
y = z_imaginary
}
printf("%1s",plot_char)
}
printf("\n")
}
exit(0)
} </lang>
Output:
# #
## ##
# ### #### #
################# ###
## # ######## ## # ## ######
### ##### ########### # #### # ##
######## ### ##### # ## ## # ## ##
#### ##### # ##### ## ##### ### #
## # ### # # ######### ## ## ######
# # #### # ########### ###### #
### ### ## # ############# #
# # ######### #### #
# ### #####
##
###
#
C++
<lang cpp>
- include <windows.h>
- include <string>
- include <complex>
const int BMP_SIZE = 600, ITERATIONS = 512; const long double FCT = 2.85, hFCT = FCT / 2.0;
class myBitmap { public:
myBitmap() : pen( NULL ), brush( NULL ), clr( 0 ), wid( 1 ) {}
~myBitmap() {
DeleteObject( pen ); DeleteObject( brush );
DeleteDC( hdc ); DeleteObject( bmp );
}
bool create( int w, int h ) {
BITMAPINFO bi;
ZeroMemory( &bi, sizeof( bi ) );
bi.bmiHeader.biSize = sizeof( bi.bmiHeader );
bi.bmiHeader.biBitCount = sizeof( DWORD ) * 8;
bi.bmiHeader.biCompression = BI_RGB;
bi.bmiHeader.biPlanes = 1;
bi.bmiHeader.biWidth = w;
bi.bmiHeader.biHeight = -h;
HDC dc = GetDC( GetConsoleWindow() );
bmp = CreateDIBSection( dc, &bi, DIB_RGB_COLORS, &pBits, NULL, 0 );
if( !bmp ) return false;
hdc = CreateCompatibleDC( dc );
SelectObject( hdc, bmp );
ReleaseDC( GetConsoleWindow(), dc );
width = w; height = h;
return true;
}
void clear( BYTE clr = 0 ) {
memset( pBits, clr, width * height * sizeof( DWORD ) );
}
void setBrushColor( DWORD bClr ) {
if( brush ) DeleteObject( brush );
brush = CreateSolidBrush( bClr );
SelectObject( hdc, brush );
}
void setPenColor( DWORD c ) {
clr = c; createPen();
}
void setPenWidth( int w ) {
wid = w; createPen();
}
void saveBitmap( std::string path ) {
BITMAPFILEHEADER fileheader;
BITMAPINFO infoheader;
BITMAP bitmap;
DWORD wb;
GetObject( bmp, sizeof( bitmap ), &bitmap );
DWORD* dwpBits = new DWORD[bitmap.bmWidth * bitmap.bmHeight];
ZeroMemory( dwpBits, bitmap.bmWidth * bitmap.bmHeight * sizeof( DWORD ) );
ZeroMemory( &infoheader, sizeof( BITMAPINFO ) );
ZeroMemory( &fileheader, sizeof( BITMAPFILEHEADER ) );
infoheader.bmiHeader.biBitCount = sizeof( DWORD ) * 8;
infoheader.bmiHeader.biCompression = BI_RGB;
infoheader.bmiHeader.biPlanes = 1;
infoheader.bmiHeader.biSize = sizeof( infoheader.bmiHeader );
infoheader.bmiHeader.biHeight = bitmap.bmHeight;
infoheader.bmiHeader.biWidth = bitmap.bmWidth;
infoheader.bmiHeader.biSizeImage = bitmap.bmWidth * bitmap.bmHeight * sizeof( DWORD );
fileheader.bfType = 0x4D42;
fileheader.bfOffBits = sizeof( infoheader.bmiHeader ) + sizeof( BITMAPFILEHEADER );
fileheader.bfSize = fileheader.bfOffBits + infoheader.bmiHeader.biSizeImage;
GetDIBits( hdc, bmp, 0, height, ( LPVOID )dwpBits, &infoheader, DIB_RGB_COLORS );
HANDLE file = CreateFile( path.c_str(), GENERIC_WRITE, 0, NULL, CREATE_ALWAYS,
FILE_ATTRIBUTE_NORMAL, NULL );
WriteFile( file, &fileheader, sizeof( BITMAPFILEHEADER ), &wb, NULL );
WriteFile( file, &infoheader.bmiHeader, sizeof( infoheader.bmiHeader ), &wb, NULL );
WriteFile( file, dwpBits, bitmap.bmWidth * bitmap.bmHeight * 4, &wb, NULL );
CloseHandle( file );
delete [] dwpBits;
}
HDC getDC() const { return hdc; }
int getWidth() const { return width; }
int getHeight() const { return height; }
DWORD* bits() const { return ( DWORD* )pBits; }
private:
void createPen() {
if( pen ) DeleteObject( pen );
pen = CreatePen( PS_SOLID, wid, clr );
SelectObject( hdc, pen );
}
HBITMAP bmp; HDC hdc;
HPEN pen; HBRUSH brush;
void *pBits; int width, height, wid;
DWORD clr;
}; class julia { public:
void draw( std::complex<long double> k ) {
bmp.create( BMP_SIZE, BMP_SIZE );
DWORD* bits = bmp.bits();
int res, pos;
std::complex<long double> c, factor( FCT / BMP_SIZE, FCT / BMP_SIZE ) ;
for( int y = 0; y < BMP_SIZE; y++ ) {
pos = y * BMP_SIZE;
c.imag( ( factor.imag() * y ) + -hFCT );
for( int x = 0; x < BMP_SIZE; x++ ) {
c.real( factor.real() * x + -hFCT );
res = inSet( c, k );
if( res ) {
int n_res = res % 255;
if( res < ( ITERATIONS >> 1 ) ) res = RGB( n_res << 2, n_res << 3, n_res << 4 );
else res = RGB( n_res << 4, n_res << 2, n_res << 5 );
}
bits[pos++] = res;
}
}
bmp.saveBitmap( "./js.bmp" );
}
private:
int inSet( std::complex<long double> z, std::complex<long double> c ) {
long double dist;//, three = 3.0;
for( int ec = 0; ec < ITERATIONS; ec++ ) {
z = z * z; z = z + c;
dist = ( z.imag() * z.imag() ) + ( z.real() * z.real() );
if( dist > 3 ) return( ec );
}
return 0;
}
myBitmap bmp;
}; int main( int argc, char* argv[] ) {
std::complex<long double> c; long double factor = FCT / BMP_SIZE; c.imag( ( factor * 184 ) + -1.4 ); c.real( ( factor * 307 ) + -2.0 ); julia j; j.draw( c ); return 0;
} </lang>
COBOL
Plots—in ASCII or EBCDIC art—a Julia set for the function f(z) = z2 + c, based on a value of c input by the user (real part then imaginary part, pressing the carriage return key after each). The sample output is for the inputs -0.8 and 0.156. <lang cobol>IDENTIFICATION DIVISION. PROGRAM-ID. JULIA-SET-PROGRAM. DATA DIVISION. WORKING-STORAGE SECTION. 01 WS-COMPLEX-CONSTANT.
05 C-REAL PIC S9V999. 05 C-IMAGINARY PIC S9V999.
01 WS-ARGAND-PLANE.
05 X PIC S9(9)V999. 05 Y PIC S9(9)V999.
01 WS-COMPLEX-VARIABLE.
05 Z-REAL PIC S9(9)V999. 05 Z-IMAGINARY PIC S9(9)V999.
01 WS-TEMPORARY-RESULTS.
05 X-SQUARED PIC S9(9)V999. 05 Y-SQUARED PIC S9(9)V999. 05 X-TIMES-Y PIC S9(9)V999. 05 Z-REAL-SQUARED PIC S9(9)V999.
01 WS-LOOP-COUNTERS.
05 HORIZONTAL PIC 999. 05 VERTICAL PIC 999. 05 ITERATIONS PIC 99.
77 WS-PLOT-CHARACTER PIC X. PROCEDURE DIVISION. INPUT-COMPLEX-CONSTANT-PARAGRAPH.
ACCEPT C-REAL FROM CONSOLE. ACCEPT C-IMAGINARY FROM CONSOLE.
CONTROL-PARAGRAPH.
PERFORM OUTER-LOOP-PARAGRAPH VARYING VERTICAL FROM 1 BY 10 UNTIL VERTICAL IS GREATER THAN 320. STOP RUN.
OUTER-LOOP-PARAGRAPH.
PERFORM COMPUTATION-PARAGRAPH VARYING HORIZONTAL FROM 1 BY 10 UNTIL HORIZONTAL IS GREATER THAN 560. DISPLAY UPON CONSOLE.
COMPUTATION-PARAGRAPH.
SUBTRACT 280 FROM HORIZONTAL GIVING X. SUBTRACT 160 FROM VERTICAL GIVING Y. DIVIDE X BY 200 GIVING X. DIVIDE Y BY 100 GIVING Y. MOVE '#' TO WS-PLOT-CHARACTER. PERFORM COMPLEX-MULTIPLICATION-PARAGRAPH VARYING ITERATIONS FROM 1 BY 1 UNTIL ITERATIONS IS GREATER THAN 50 OR WS-PLOT-CHARACTER IS EQUAL TO SPACE. DISPLAY WS-PLOT-CHARACTER UPON CONSOLE WITH NO ADVANCING.
COMPLEX-MULTIPLICATION-PARAGRAPH.
MULTIPLY X BY X GIVING X-SQUARED. MULTIPLY Y BY Y GIVING Y-SQUARED. SUBTRACT Y-SQUARED FROM X-SQUARED GIVING Z-REAL. ADD C-REAL TO Z-REAL. MULTIPLY X BY Y GIVING X-TIMES-Y. MULTIPLY X-TIMES-Y BY 2 GIVING Z-IMAGINARY. ADD C-IMAGINARY TO Z-IMAGINARY. MULTIPLY Z-REAL BY Z-REAL GIVING Z-REAL-SQUARED. IF Z-REAL-SQUARED IS GREATER THAN 10000 THEN MOVE SPACE TO WS-PLOT-CHARACTER. MOVE Z-REAL TO X. MOVE Z-IMAGINARY TO Y.</lang>
- Output:
###
## ##
# ### #### #
################ ###
# # # ############### # #######
## ##### ######## # # ##### ##
######## ### ######## # ### ### # ## #
## # #### # # # #### # #### ### #
# # ### # # ######### # ## ######
# ######### # ######### ###### #
### ### ## ## ############ #
# # ######### ## ## #
##### ####
# #
###
#
J
<lang j>load '~addons/graphics/fvj4/complex_dynamics.ijs' pal2=: 255,~0,<.(254$1 0.8 0.6)*Hue 5r6*(i.%<:)254 g=: [: %: 0.3746j0.102863 0.132565j0.389103 _0.373935j_0.353777 1&p. view_image pal2;b=:g escapetc (10 255) 500 zl_clur _1.5 1.5j1.5</lang>
See also: Fractals Visualization and J, 4th edition, Part 1 (by Clifford A. Reiter), Chapter 6
See http://webbox.lafayette.edu/~reiterc/mvp/ec_julia/index.html for some other examples. (That said, note that this is a link into a small college site and it might drift over time. In the past, for example, you would have had to use 'www' where it currently says 'webbox')
Java
<lang java>import java.awt.*; import java.awt.image.BufferedImage; import javax.swing.*;
public class JuliaSet extends JPanel {
private final int maxIter = 300; private final double zoom = 1; private double cY, cX;
public JuliaSet() {
setPreferredSize(new Dimension(800, 600));
setBackground(Color.white);
}
void drawJuliaSet(Graphics2D g) {
int w = getWidth();
int h = getHeight();
BufferedImage image = new BufferedImage(w, h,
BufferedImage.TYPE_INT_RGB);
cX = -0.7;
cY = 0.27015;
double moveX = 0, moveY = 0;
double zx, zy;
for (int x = 0; x < w; x++) {
for (int y = 0; y < h; y++) {
zx = 1.5 * (x - w / 2) / (0.5 * zoom * w) + moveX;
zy = (y - h / 2) / (0.5 * zoom * h) + moveY;
float i = maxIter;
while (zx * zx + zy * zy < 4 && i > 0) {
double tmp = zx * zx - zy * zy + cX;
zy = 2.0 * zx * zy + cY;
zx = tmp;
i--;
}
int c = Color.HSBtoRGB((maxIter / i) % 1, 1, i > 0 ? 1 : 0);
image.setRGB(x, y, c);
}
}
g.drawImage(image, 0, 0, null);
}
@Override
public void paintComponent(Graphics gg) {
super.paintComponent(gg);
Graphics2D g = (Graphics2D) gg;
g.setRenderingHint(RenderingHints.KEY_ANTIALIASING,
RenderingHints.VALUE_ANTIALIAS_ON);
drawJuliaSet(g);
}
public static void main(String[] args) {
SwingUtilities.invokeLater(() -> {
JFrame f = new JFrame();
f.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
f.setTitle("Julia Set");
f.setResizable(false);
f.add(new JuliaSet(), BorderLayout.CENTER);
f.pack();
f.setLocationRelativeTo(null);
f.setVisible(true);
});
}
}</lang>
Julia
<lang julia>using Images
w, h = 800, 600 img = Array(UInt8, h, w, 3)
maxIter = 50 c = -0.8+0.156im
function hsv2rgb(h, s, v)
c = v * s x = c * (1 - abs(((h/60) % 2) - 1)) m = v - c
r,g,b =
if h < 60
(c, x, 0)
elseif h < 120
(x, c, 0)
elseif h < 180
(0, c, x)
elseif h < 240
(0, x, c)
elseif h < 300
(x, 0, c)
else
(c, 0, x)
end
r = round(UInt8, (r + m) * 255) g = round(UInt8, (g + m) * 255) b = round(UInt8, (b + m) * 255)
b,g,r
end
for x in 1:w
for y in 1:h
i = maxIter
z = Complex((x - w/2) / w * 3, (y - h/2) / h * 2)
while abs(z) < 2 && (i -= 1) > 0
z = z*z + c
end
r,g,b = hsv2rgb(i / maxIter * 360, 1, i > 1 ? 1 : 0)
img[y,x,1] = r
img[y,x,2] = g
img[y,x,3] = b
end
end
save("JuliaSet.png", colorim(img, "RGB"))</lang>
Perl
<lang perl>use Imager;
my($w, $h, $zoom) = (800, 600, 1); my $img = Imager->new(xsize => $w, ysize => $h, channels => 3);
my $maxIter = 255; my ($cX, $cY) = (-0.7, 0.27015); my ($moveX, $moveY) = (0, 0);
my $color = Imager::Color->new('#000000');
foreach my $x (0 .. $w - 1) {
foreach my $y (0 .. $h - 1) {
my $zx = (1.5 * ($x - $w / 2) / (0.5 * $zoom * $w) + $moveX);
my $zy = (($y - $h / 2) / (0.5 * $zoom * $h) + $moveY);
my $i = $maxIter;
while ($zx**2 + $zy**2 < 4 and --$i >= 0) {
($zy, $zx) = (2 * $zx * $zy + $cY, $zx**2 - $zy**2 + $cX);
}
$color->set(hsv => [$i / $maxIter * 360, 1, $i > 0 ? 1 : 0]);
$img->setpixel(x => $x, y => $y, color => $color);
}
}
$img->write(file => 'julia_set.png');</lang>
Perl 6
with the pallette swapped, just because.
<lang perl6>use Image::PNG::Portable;
my ($w, $h) = 800, 600; my $out = Image::PNG::Portable.new: :width($w), :height($h);
my $maxIter = 255; my $c = -0.7 + 0.27015i;
julia($out);
$out.write: 'Julia-set-perl6.png';
sub julia ( $png ) {
for ^$w -> $x {
for ^$h -> $y {
my $z = Complex.new(($x - $w / 2) / $w * 3, ($y - $h / 2) / $h * 2);
my $i = $maxIter;
while (abs($z) < 2 and --$i) {
$z = $z*$z + $c;
}
$png.set: $x, $y, |hsv2rgb($i / $maxIter * 360, 1, ?$i).reverse;
}
}
}
sub hsv2rgb ( $h, $s, $v ){
my $c = $v * $s;
my $x = $c * (1 - abs( (($h/60) % 2) - 1 ) );
my $m = $v - $c;
my ($r, $g, $b) = do given $h {
when 0..^60 { $c, $x, 0 }
when 60..^120 { $x, $c, 0 }
when 120..^180 { 0, $c, $x }
when 180..^240 { 0, $x, $c }
when 240..^300 { $x, 0, $c }
when 300..^360 { $c, 0, $x }
}
( $r, $g, $b ) = map { (($_+$m) * 255).Int }, $r, $g, $b;
}</lang>
Sidef
<lang ruby>require('Imager')
var (w, h) = (640, 480) var img = %s'Imager'.new(xsize => w, ysize => h, channels => 3)
var maxIter = 50 var c = Complex(-0.388, 0.613)
var color = %s'Imager::Color'.new('#000000')
for x,y in (^w ~X ^h) {
var i = maxIter
var z = Complex((x - w/2) / w * 3, (y - h/2) / h * 2)
while (z.abs < 2 && --i) {
z = (z*z + c)
}
color.set(hsv => [i / maxIter * 360, 1, i])
img.setpixel(x => x, y => y, color => color)
}
img.write(file => "JuliaSet_sidef.png")</lang>
zkl
Uses the PPM class from http://rosettacode.org/wiki/Bitmap/Bresenham%27s_line_algorithm#zkl
<lang zkl>fcn juliaSet{
w,h,zoom:=800,600, 1; bitmap:=PPM(w,h,0xFF|FF|FF); // White background
cX,cY:=-0.7, 0.27015; moveX,moveY:=0.0, 0.0; maxIter:=255;
foreach x,y in (w,h){
zx:=1.5*(x - w/2)/(0.5*zoom*w) + moveX;
zy:=1.0*(y - h/2)/(0.5*zoom*h) + moveY;
i:=maxIter;
while(zx*zx + zy*zy < 4 and i > 1){
tmp:=zx*zx - zy*zy + cX; zy,zx=2.0*zx*zy + cY, tmp; i-=1;
}
// convert byte to RGB (3 bytes), kinda magic to get nice colors
bitmap[x,y]=i.shiftLeft(21) + i.shiftLeft(10) + i*8;
}
bitmap.writeJPGFile("juliaSet.jpg",True);
}();</lang>