1. L-Systems Lindenmayer Systems Prusinkiewicz and Lindenmayer
Recursive branching structure
Complex but constrained.

2. Recursive Branching Structure

3. Botany - structure Stems, roots, buds, leaves, flowers
Herbaceous or woody
Dichotomous or monopodial
Terminal bud or lateral bud
Vegetative bud or flower bud
Alernate, opposite, or whorled.

4. Botany - growth influences
Lineage - age
Cellular descent - nutrients and hormones
Tropisms - e.g., phototropism, geotropism
Obstacles - collision response: temporary or permanent.

5. DOL-Systems Production rules: ai -> bi D - deterministic
0 - zero contextual systems (context-free)
Axiom - initial string
Parallel rewriting rules.

6. Geometric Interpretation of String
Geometric replacement
- replace each symbol of string with a graphic element
- connectivity assumptions
Turtle graphics
- each symbol is a command to a drawing pen.

7. Geometric Replacement - Example
A -> BCB
B -> XX
C -> TT
Axiom: A A
BCB
XXTTXX

8. Geometric Replacement
XXTTXX X: T: X X T T X X

9. Turtle Graphics “drawing turtle” is controlled by symbols of string
Symbols read from string in left to right order
State of turtle: position and direction: (x, y, a)
Parameters
- d: linear step size
- d: angular step size.

10. Turtle Graphics F
Move forward a distance d while drawing a line. Its state will change from (x,y,a) to (x+d*cos a, y+d*sin a, a) f
Move forward a distance d without drawing a line. Its state will change as above. +
Turn left by an angle d. Its state will change from (x,y, a) to (x,y, a + d) -
Turn right by an angle d. Its state will change from (x,y, a) to (x,y, a - d).

11. Turtle Graphics - Example
A -> BCB
B -> FF
C -> TT
T -> -F++F-
Axiom: A
BCB
FFTTFF
FF -F++F- -F++F- FF

12. Turtle Graphics - Exampled:
d: 45 degrees
Initial state: (10,10,0)
FF -F++F- -F++F- FF F F
-F++F-
-F++F- F F

13. Bracketed L-Systems Brackets - push and pop states
- multiple types of brackets
- brackets have attachment semantics
X X [ XX ] { XX } X X

14. Properties Can be recursive - reduce size based on branching level
- self-similarity under scale
Attribute symbols: line length, width, color, etc.
Database amplification
Track string development => growth process.

20. Non-Deterministic S -> F A F A -> [ + F B F ] F F F
F [ + F F F ] F
F F F
S -> F A F
A -> [ + F B F ]
A -> F
B -> [ - F B F ]
B -> F
Axiom: S
F [ + F [ - F F F ] F ] F

22. Stochastic L-Systems S 1.0 -> F A F A 0.8 -> [ + F B F ]
B 0.4 -> [ - F B F ]
B 0.6 -> F

23. Context-Sensitive L-Systems
S -> F A T
A>T -> [ + F B F ]
A>F -> F
F<B -> [ - F A F ]
F<B >F -> [ - F A F ]
T -> F

24. Animating Plant Growth
Flexible movement of structure
Changes in topology due to growth process
Elongation of existing structures
F -> FF
F -> F1
F1 -> F2
F2 -> F3
F 3-> F4

25. Parametric L-Systems S -> A(0) A(t) -> A(t+1)
A(t): t>=1.0 -> F

26. Timed L-Systems Global time variable Local age value
(u0, b0) -> ( (u1,a1), (u2, a2), … , (un, an) )
Jump from terminal age to terminal age.

27. Interacting with the Environment
Physical obstacles
Wind
Gravity
Nutrients, moisture from soil
Sunlight: length of day, shadows.

28. Open L-Systems Mech and Prusinkiewicz Communication terms
- Request and transmit information w/ environment
- Distribution of nutrients, sunlight, etc.

29. L-Systems - Summary Branching structure, self-similarity
Controlled complexity , data base amplification
Geometric replacement, Turtle graphics
Non-deterministic, stochastic, context sensitive, timed, open
Animation
- dynamics of static structure
- time-varying drawing parameters
- topology changes
- growth of individual elements.