• Welcome to TechPowerUp Forums, Guest! Please check out our forum guidelines for info related to our community.

Fractal Generator FAQ and How-to

Oliver_FF

New Member
Joined
Oct 15, 2006
Messages
544 (0.13/day)
Likes
65
Processor Intel q9400 @ stock
Motherboard Lanparty P45-T2RS
Cooling Zalman CNPS-9500
Memory 8GB OCZ PC2-6400
Video Card(s) BFG Nvidia GTX285 OC
Storage 1TB, 500GB, 500GB
Display(s) 20" Samsung T200HD
Case Antec Mini P180
Audio Device(s) Sound Blaster X-Fi Elite Pro
Power Supply 700w Hiper
Software Ubuntu x64 virtualising Vista
#1
Ok, so how many of you have heard of a Fractal Generator? XD When I first created my own fractal generator from scratch I did some reading about it, and nothing was really laid out simply enough for me to fully understand it with really basic maths understanding, so i'm gonna attempt to correct this :toast:


What on earth is a fractal??
A fractal is an Infinitely Complex shape?
A little maths background
Sorry guys, a little maths has to come here. Suppose you have a quadratic equation (something that looks like this:

x^2 + 3x - 2 and you want to find it's "roots" (the places where the line crosses the X-axis. You --could-- solve this by solving the following X^2 +3x -2 = 0, but what if the equation is REALLY tough?

One way of doing this is to guess a number, and put it through the equation. Take the new result, put it in again. Take the new result and put it in again. Eventually your number will either run up to infinity or settle to one specific number. If it runs up and up, it's said to tend towards infinity.

The next part of maths you need to be aware of are "Imaginary Numbers" :rockout:
In particular, you need to know all about the number "i". i carries the value of squareroot(-1). Now, don't try to work out it's value, because it's impossible to squareroot a negative number and you'll just get errors on your calculators. The important thing about i is that i^2 = -1

A complex number in my context is something that looks like 3i + 5. It basically has two parts, a real number and an imaginary number.

So where's this going??
Well, a Fractal is drawn by using the method of solving a quadratic equation i mentioned earlier. You start by taking a massive grid, I've chosen to use 1024x768 pixels, as it's a cool number :cool:

You then imagine a graphs axis running through the middle your image, as if your looking at some graph paper from a science experiment back at school.

Along the grid from left to right is the x-axis. The line from the top to the bottom is the i axis.

So what's a Fractal? It's a visual representation of which of your grid locations tend off towards infinity when you put them through any mathematical equation XD


The Mandelbrot Set
Click for a Preview!!!
This fractal is generated by using the equation x^2 + x. You take a co-ordinate in the form of a complex number, ie. for co-ordinate (1, 1) you have the complex number (1i + 1). You stick this number into the equation and you get a new number out. You put this new number into the equation and get a new number out... keep repeating. If the value you get out goes over say 4, then it's safe to assume that it has tended towards infinity. When this happens you can assign that specific co-ordinate a colour depending on how many times you've repeated the equation.

Using (1i + 1):
(1i + 1)^2 + (1i + 1) --> expand the first part
(1i + 1)(1i + 1) + (1i + 1) --> multiply out the bracket
(1i^2 + 2i + 1) + (1i + 1) --> simplify i^2 for -1
(-1 + 2i + 1) + (1i + 1) --> add the two together
(3i + 1)
the numbers 1 and 3 are both below 4, so we put (3i + 1) back into the formula.

Now, this isn't very easy to program, so i'll be nice and work out a scenario algebraically.
Suppose our co-ordinate is (ai + b) where a is the y co-ordinate and b is the x co-ordinate.

Putting that into the equation gives:
(ai + b)^2 + (ai + b)
(ai + b)(ai + b) + (ai + b)
((a^2)(i^2) + (2bai) + (b^2)) + (ai + b)
((-a^2) + 2bai + (b^2) + (ai + b)
b^2 + 2bai - a^2 + ai + b

Now, separate out the components involving i:
2bai + ai --> factor the i's out
(2ba + a)i
And those that don't:
b^2 - a^2 + b

Still with me? XD
For those that lost interest, here's the important bit
So, if you take this co-ordinate:
(ai + b)
then after you apply the formula you'll have this:
((2ba + a)i + (b^2 - a^2 + b))



The Julia Set
This is basically the same as the Mandelbrot set, but instead of having the equation x^2 + x, you use:
x^2 + c, where c is the initial co-ordinate, and always is.

The Julia set is said to be a Subset of the Mandelbrot set, because if you pick a point on in the mandelbrot fractal and put those values into the Julia set, you'll get something similar but different.


My Fractal Generator
Well, it seems i can't upload my app, but i suppose that's fair enough. I've written it using C++ and the SDL libraries in the Dev-C++ IDE, but i've found someone to host the app anyways.

You can see what it looks like in the screenshots below, you'll need the SDL.dll runtime file found here:
http://www.libsdl.org/release/SDL-1.2.12-win32.zip
just stick the DLL in the same directory as the .exe and it'll work :)

And my app is here:
http://www.smutchings.com/downloads/fractal.rar

The background to the window is the Mandelbrot set, the top left window draws the Julia set with the corresponding constant relative to where your mouse is over the Mandelbrot set. The bottom left window draws the Mandelbrot set again, but you can move and zoom in infinitely to check out some of the cool features.

The Controls
Mouse cursor - alters the constant values for julia set and moves the zoom window.
Home/PageUp - increases the zoom in the zoom window.
End/PageDown - decreases the zoom in the zoom window.
Arrow Keys - fine tune the view in the zoom window.
 

Attachments

Last edited by a moderator:
Joined
Jan 12, 2007
Messages
2,276 (0.55/day)
Likes
230
Location
Cairns QLD, Aussie
System Name GaMEr / HTPC
Processor FX8320 4.5ghz so far NB- 2815mhz / X2 255 @ 3.2ghz (Temp till BD)
Motherboard 990FX-UD5 / GA-MA785GPMT-US2H
Cooling XSPC - 240rad 480rad 5 x 120mm out, 6 x 80mm in / H220-edge 1/2" & 2 x 120mm Magma
Memory 4gb PI 6,8,6,18 1T 1706mhz / 4gb Crucial D9's 1400 @ 6,6,6,18,1T
Video Card(s) 3gb HD7950 / Onboard
Storage 2 x 120gb Sandisk ultra plus 2tb WD / 60gb vertex 2 + 2 x 2tb WD Greens
Display(s) 3x24in Dell Ultra (portrait) / NEC 32" FullHD LCD
Case LL PC71A Full tower / Antec fusion black with Imon remote Modded for WC
Audio Device(s) OB / Auzentech cinema 7.1
Power Supply TT 700w TR2 / TT 600w Litepower modded with Noctua S12 fan.
Software 7 Ult / 7 premium x64
#2
Man I just woke up and this is the first thing I read....

OWWWWW.... my head.

Will try and read it again later on.. :)
 

Oliver_FF

New Member
Joined
Oct 15, 2006
Messages
544 (0.13/day)
Likes
65
Processor Intel q9400 @ stock
Motherboard Lanparty P45-T2RS
Cooling Zalman CNPS-9500
Memory 8GB OCZ PC2-6400
Video Card(s) BFG Nvidia GTX285 OC
Storage 1TB, 500GB, 500GB
Display(s) 20" Samsung T200HD
Case Antec Mini P180
Audio Device(s) Sound Blaster X-Fi Elite Pro
Power Supply 700w Hiper
Software Ubuntu x64 virtualising Vista
#3
long double a, b, c, e, f, g, h;

//void ColorPixel(x-pos, y-pos, R value, G value, B value); --> You'll need a way of plotting pixels!



//Mandelbrot set:
for (x=0; x<1024; x++){ //window size 1024x768
for (y=0; y<768; y++){
a = x-830; a = a/400; //Scale to fit the window!
b = y-400; b = b/320;
g=a;
h=b;
for (p = 0; p<255; p++){ //Loop some
c = a;
a = ((a * a) - (b * b)) + g;
b = (2 * c * b) + h;
if (a > 4) break;
if (b > 4) break;
}
PlotPixel(x,y,p*2,0,0); //Here's where you want to plot the pixel to screen!
}
}

[edit] Gah for the forum removing my indenting!
 
Joined
Jul 3, 2007
Messages
1,317 (0.33/day)
Likes
211
Location
New Zealand
Processor AMD Phenom II 555BE unlocked X4 @3.8GHz
Motherboard GA-78LMT-S2P
Cooling Thermaltake Blue Orb + open case in cold room = low temps lol
Memory 8Gb DDR3 1600
Video Card(s) GTX580
Storage 64GB SSD Super Talent
Display(s) 22" Chimei, 17" Philips
Case POS!
Power Supply Cooler Master Silent Pro Gold 800W
#5
man i dont get it. probably why i failed first year calculus! passed it 2nd time round tho
 
Joined
Aug 10, 2007
Messages
4,107 (1.04/day)
Likes
1,175
Location
Geneva, FL, USA
Processor Intel i5-6600
Motherboard ASRock H170M-ITX
Cooling Cooler Master Geminii S524
Memory G.Skill DDR4-2133 16GB (8GB x 2)
Video Card(s) Gigabyte R9-380X 4GB
Storage Samsung 950 EVO 250GB (mSATA)
Display(s) LG 29UM69G-B 2560x1080 IPS
Case Lian Li PC-Q25
Audio Device(s) Realtek ALC892
Power Supply Seasonic SS-460FL2
Mouse Logitech G700s
Keyboard Logitech G110
Software Windows 10 Pro
#6
PHP/GD:

PHP:
$imgX = 640;
$imgY = 480;

$img = imagecreatetruecolor($imgX, $imgY);

for ($x = 0; $x < $imgX; ++$x) {
	for ($y = 0; $y < $imgY; ++$y) {
		$a = $x - 830;
		$a = $a / 400;
		$b = $y - 400;
		$b = $b / 320;
		$g = $a;
		$h = $b;
		for ($p = 0; $p < 255; ++$p) {
			$c = $a;
			$a = (($a * $a) - ($b * $b)) + $g;
			$b = (2 * $c * $b) + $h;
			if ($a > 4 || $b > 4) {
				break;
			}
		}
		imagesetpixel($img, $x, $y, imagecolorallocate($img, $p, 0, 0));
	}
}

header('Content-Type: image/jpeg');
imagejpeg($img);
imagedestroy($img);
 

Oliver_FF

New Member
Joined
Oct 15, 2006
Messages
544 (0.13/day)
Likes
65
Processor Intel q9400 @ stock
Motherboard Lanparty P45-T2RS
Cooling Zalman CNPS-9500
Memory 8GB OCZ PC2-6400
Video Card(s) BFG Nvidia GTX285 OC
Storage 1TB, 500GB, 500GB
Display(s) 20" Samsung T200HD
Case Antec Mini P180
Audio Device(s) Sound Blaster X-Fi Elite Pro
Power Supply 700w Hiper
Software Ubuntu x64 virtualising Vista
#7
Haha, nicely done - didn't think of that XD

If you adjust these:
Code:
        $a = $x - 830;
        $a = $a / 400;
        $b = $y - 400;
        $b = $b / 320;
you'll be able to see more/pan the image/zoom in :eek:
 

Oliver_FF

New Member
Joined
Oct 15, 2006
Messages
544 (0.13/day)
Likes
65
Processor Intel q9400 @ stock
Motherboard Lanparty P45-T2RS
Cooling Zalman CNPS-9500
Memory 8GB OCZ PC2-6400
Video Card(s) BFG Nvidia GTX285 OC
Storage 1TB, 500GB, 500GB
Display(s) 20" Samsung T200HD
Case Antec Mini P180
Audio Device(s) Sound Blaster X-Fi Elite Pro
Power Supply 700w Hiper
Software Ubuntu x64 virtualising Vista
#8
man i dont get it. probably why i failed first year calculus! passed it 2nd time round tho
Basically, it's a 2d graph of the time it takes for a co-ordinate to tend towards infinity for the equation y=x^2 + c

You get some truly weird shapes :eek:
 
Top