The Mandelbrot set shown to the right determines another set of fractals, so
called Julia sets. Each point c in the Mandelbrot set determines a Julia set J(c).
Picking and dragging with left mouse in the Mandelbrot image on the right will
generate a new Julia set J(c) shown to the left. Mathematically we can say that
the Mandelbrot set is the parameter space which describes the entirety of Julia
sets.
For a given parameter c the Julia set J(c) is defined as the subset of points
z of the complex plane for which the iteration
Iterate[z -> z^2+c]
does not diverge to infinity. Computationally, the
algorithm checks for each pixel z if the sequence z,Iterate(z),Iterate(Iterate((z)),...
stays within a certain region of the complex plane after a fixed number of
iterations. The number of performed iterations can be adjusted using the slider
"Number of iterations". The slider "Block Size" allows to work with a coarser
pixel grid to speed up computations at the cost of resolution.
When the parameter c lies inside the black region of
the Mandelbrot set then the corresponding Julia set has a non-empty black
interior.
Some possible interactions:
- Pick inside the Mandelbrot image and see the Julia set being recomputed.
The components of the picked complex parameter c is shown.
- Pick inside the Julia image and see the iterates of the picked point
under the Julia map.
- Select a rectangular region in display to zoom into image: keep pressing
the key 'm' while dragging a rectangle.
- Press the RESET button to initialize the applet,
and to zoom out.
- The HUE slider changes the base value of the rainbow colors.
Note, this is applet is mainly a tutorial for studying image handling in JavaView
rather than a full-fledged application to study fractals.
View source code:
PaJuliaSet.java of applet
PjJuliaSet.java of project
PjJuliaSet_IP.java of info panel.
