Peano curves
Space filling curves (completely
fill a region of space )
Have fractal dimension of 2
Ex.
Hilbert curves, Sierpinski curves
Polya’sPeano curve
Generated
by replacing each segment of a generation by a right angled elbow
Direction
of the elbow alternate in a L , LR , LRLR ,… fashion
To save
the current state and to restore characters ‘]’
and ‘]’ are added to the language
L –
Systems
Approach
to generate curves
Generate
curves based on simple set of rules ( productions)
String
Production rules
F àF – F ++ F – F
F means
forward (1,1)
+ means
turn(A)
- means
turn(-A)
à Means that every F is replaced
by F – F ++ F – F
Atom –
initial string
Production rules are applied to atom to
produce the first generation string (S1 )
Generation
of richer set of curves
by adding
more rules to string production process richer set of curves can be generated
Ex.
Dragon curves
F àF
X àX + YF +
Y à – FX – Y
Atom = FX
Order1
String S1 Order2
String S2
Atom = FX
S1 = FX
+ YF + S2 = FX
+ YF ++ – FX – YF +
To generate the second generation string
the same production is applied to the
first generation string
