An Important Class of Fractals
One of important classes of fractals is represented by parameterized curves
A continuous curve definition was appeared in J.Hutchinson's paper
"Fractals and Self Similarity"
Iterated Function System
is called a fractal parameterized curve
if the following condition is satisfied:
for all i=1,..,n-1,
where x1 and xn is fixed points of according maps
S1 and Sn.
Hutchinson proved that for any system of contraction,
with such condition, there exist a continuous map f
from [0,1] segment
to F, such that f( [0,1] ) = F.
The most known fractal curves are: Koch curve, Levy curve, Peano curve
and Sierpinsky triangle.
Regrettably Dragon set does not satisfy Hutchinson's fractal curve definition.
The following Java applet can draw some continuous curves which approximates
a curves and sets considered above.
Fractal microscope snowflake
is alternative curve drawing applet.
Although it does not draw such curves as Sierpinsky Triangle
and Dragon set but it allows to modify maps of curve.
Due to such freedom you can draw your own curves.
- With my IFS editor applet you can create
explore and modify all sets considered above and other