Peano curves

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