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The Mandelbrot set is fun but implementing a simple viewer in Silverlight can be a challenge. The reason is that Silverlight is primarily a vector graphics system and the natural way to think of plotting the Mandelbrot set is to plot pixels. But in a web based application that doesn't have the advantage of hardware graphics acceleration this isn't a good way to do the job. Instead we can make use of Silverlight's bitmap object to create the image and then present it to the user after it has been precomputed.
This project is based on Mandelbrot Zoomer in WPF and comparing the two gives some insight into the differences between WPF and Silverlight.
You can see a live demonstration of the Zoomer if you go to the end of this article. You will need Silverlight 4 installed to see it work.
If you would like some background reading on fractals and the Mandelbrot set in particular then see - Fractals
In this project we plot the Mandelbrot set and allow the user to zoom in on any area of interest. The zooming is done "live" in the sense that the area is recomputed rather than simply zooming in to a pre-computed image.
The general programming topics covered in this project include:
The Mandelbrot set
The actual Mandelbrot computation is very easy.
In fact it is this very
simplicity that makes it so amazing that something so complex is the end product.
If you consider a small 2D region centred on (0,0) i.e. the origin, then a point (a,b) is in the Mandelbrot set if the iteration:
does not diverge to infinity where c=a+ib and z is a general complex number of the form x=x+iy. (Recall that i is a number such that i2=-1).
To compute the Mandelbrot set all we do is start off with a number c and an initial value for z, usually 0, and then replace z by its square plus c.
This operation is repeated, generating a sequence of z values. Either z gets bigger and bigger, in the sense of moving away from the origin, or it doesn't. If it doesn't then it is in the Mandelbrot set.
Until .NET 4.0 our next problem would have been finding a good way of doing complex arithmetic in C#. However, in .NET 4.0 a Complex type was introduced and we can now simply create complex variables and do complex arithmetic more or less as written in the usual notation. For more details see Not so complex numbers in C#.
The only real problem concerns deciding when the iteration has resulted in a value which is going off to infinity and when it has not. B
asically what we do is iterate a large number of times and if by then the value has not wandered outside of the circle of radius 2 centred on the origin then it probably isn’t going to wander off to infinity. (It can be proved that if any point does go outside of this circle then is most certainly does go on to infinity.)
Putting all this together we have the following algorithm:
- Set c to the x,y co-ordinates of the point you want to test
- Set z to zero
- Set count to zero
- Using the current values work out z=z2+c
- if mod(z)>2 then point goes to infinity and is not part of the Mandelbrot set
- Add one to count if it’s >1000 then the point is still close to the origin after 1000 iterations and is part of the Mandelbrot set
- Repeat by going back to step 4 unless you have determined whether or not the point is in the Mandelbrot set.
Translating this into a function is fairly easy:
private Int32 mandelbrot(Complex c)
Int32 count = 0;
Complex z = Complex.Zero;
while (count < 1000 && z.Magnitude < 4)
z = z * z + c;
Start a new Silverlight 4 application called SilverZoomer. You can use full Visual Studio or Visual Web Developer 2010 Express - both work equally well. Add the above function to the
To make this work we have to load a reference to the System.Numerics assembly. Right click on References in the Project Explorer window and select Add References. Navigate to System.Numerics. Also add:
to the start of the program.
There’s very little to add to the earlier comments about this function except the fact that it returns the value of count. If count equals 1000 then the point is in the Mandelbrot set and any other value gives you a measure of how quickly the point goes off to infinity.
The value of count is often used to color the non-Mandelbrot points according to how fast they shoot off to infinity and we will return to this topic when we get to plotting the set.