Fractals -8- Dr Christoph Traxler.pdf

8/14/2019 Fractals -8- Dr Christoph Traxler.pdf

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8. L-Systems

8.1

L-Systems

Christoph Traxler 1

L-Systems

Christoph Traxler 2


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8. L-Systems

8.2

Christoph Traxler 3

Christoph Traxler 4


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8. L-Systems

8.3

L-Systems

Introduced 1968 by the biologistAristid Lindenmayer as a

development of simplemulticellular organisms

Subsequently applied to investigate higherplants and plant organs

Christoph Traxler 5

L-systems initialy generated the topology ofplants as a result of discrete development

L-Systems

Prusinkiewicz extendedL-systems in various ways:

-

Timed L-systems

Open L-systemsSimulation of ecosystems

Christoph Traxler 6

allows detailed modeling and a realisticvisualization of plants


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8. L-Systems

8.4

L-Systems

L-systems are parallel rewriting systemsoperating on strings of symbols

- , , where

A is a finite set of symbols called alphabet

A+ is a non empty word called axiom

*

Christoph Traxler 7

s a n e se o pro uc ons

L-Systems

A production is written as av and meansthat each occurence of the symbol a in the

In a derivation step all symbols of the

current string are replaced simultaneouslyby the proper production

Difference compared with Chomsky-systems:

Christoph Traxler 8

Only one symbol is replaced in eachderivation step


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8. L-Systems

8.5

L-Systems

If there exists no production for a symbol athen the identic production aa is applied

an L-system is called formal language of theL-system

An L-system is called deterministic (dL-system) if and only if there exists for each a

Christoph Traxler 9

A exactly one vA* so that av

L-Systems

Example:{a,b,c};a;{p1:aab,p2:bac}0 = a

1 = ab

2 = ab ac

3 = abacabc

4 = abacabcabacc

Christoph Traxler 10

ow are er ve str ngs trans orme nto agraphic ?


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8. L-Systems

8.6

Turtle Graphics

The turtle is a drawing tool for the geometricinterpretation of a derived string

specific commands for certain symbolsand changes its state

Its state is defined by the triplet (x,y,),where x,y defines its position on the plane


and the angle its orientation

Turtle Graphics

The turtle obeys to the following commands:

F draw a line of length d, the state

changes to (x+d cos, y+d sin, )

f move forward a step of length d without

drawing+ turn left by angle , the state changes to

x +


- turn right by angle , the state changes

to (x,y,-)


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8. L-Systems

8.7

Turtle Graphics

Starting conditions for the turtle:Initial state, d and

- - - - - ,d=1, =90

F


+ -start

Classic Fractals by L-Systems

Using the turtle commands as alphabet

Construction of Kochs island:

= , epen s on er va on eng

= F--F--F //Initiator

F F+F--F+F //Generator



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8. L-Systems

8.9


Space Filling Curves

Node Replacement: Using additional symbols

derivation

Construction of the Hilbert curve


L+RF-LFL-FR+ R-LF+RFR+FL-:L


Hilbert curve



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8. L-Systems

8.10

Branching Structures

The turtle constructs objects incrementaly,thus there must be a mechanism to move it

This is achieved by additional commands:

[ push the state of the turtle onto a stack

] pop the last state from the stack and

make it to the current state of the turtle


Can be used to store and restore otherinformation as well


2nd order branchterminal node

branching point

1st order branch


F[+F][-F[-F]F]F[+F[+F][-F][-F]

segment


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8. L-Systems

8.11


n=5,=25.7:FFF[+F]F[-F]F


n=5,=20:FFF[+F]F[-F][F]


Node replacement

n=7,=20

:X F[+X]F[-X]+XF FF


n=5,=22.5

:XX F-[[X]+X]+F[+FX]-XF FF


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8. L-Systems

8.12

Non Deterministic L-Systems

At least one symbol has more than oneproduction

one of the productions for each symbolduring derivation:

Stochastic L-systems

Context sensitive L-systems


Parametric L-systems

Stochastic L-Systems

Different productions for a symbol areselected randomly

number is used for each symbol

A probability is assigned to each productionUsed to generate variations among individualsof a s ecies



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8. L-Systems

8.13


: Fp1: F

F[+F]F[-F]F: F F +F F

p3: F F[-F]F





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8. L-Systems

8.14

Context Sensitive L-Systems

The selection of a production for a symboldepends on the adjacent symbols in the

A context-sensitive production is writtenasA*A* A+, example:

xy < a > z ab


xyazabyxayzab xyabzabyxaayzab


Is used to simulate the propagation of signals(hormones, nutrients) between parts of a plant

: baaaaaaap1:b


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8. L-Systems

8.15


Branching structures: Topology has to beconsidered when searching for the context

gnore -

: Fb[+Fa]Fa[-Fa]F[+Fa]Fap: Fb< Fa Fb


Parametric L-Systems

Symbols are assocciated with a finite set ofparameters

productions and to controll turtle geometry

Incorporation of geometrical features, thefinal shape is a result of the derivation

Im ortant for lant modelin because the


geometry of a plant is the result of itsdevelopmental process


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8. L-Systems

8.16


A parametric L-system (pL-system) is definedas ordered quadruplet A,,,P, where

is a finite set of parameters

(A R*)+ is the axiom

P ((A*) C() (A E())* is the set ofproductions


C() denotes a logical and E() an arithmeticexpression with parameters from


PL-systems operate on strings of moduleswhich are parametric symbols, i.e. (AR*)

er va on: pro uc on p ma c es a mo u em of a string if:

The symbols of m and of the predecessorof p are the same

The number of real values in m is equal to


the number of parameters in thepredecessor of p

The condition of p is true


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8. L-Systems

8.17


Modules are replaced as follows:

1. All parameters in the production are set to

2. The condition is evaluated

3. If the condition is true, then the arithmeticexpressions in the successor are evaluated

4. The new modules with the resulting real


values are filled into the new string


: B(2)A(4,4)p1: A(x,y): y3 B(x)A(x/y,0)p3: B(x) : x=1 B(x-1)

Result: B(2)A(4,4)B(1)B(4)A(1,0)


,

C B(2)A(4,3)

C B(1)A(8,7)

C B(0)B(8)A(1.142,0)


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8. L-Systems

8.18

Virtual construction tool in space

Its state is defined by a local coordinate

The 3D-Turtle

Within pL-systems the commands dependon parameters

head (h-axis)


up u-ax s

right (r-axis)

The 3D-Turtle Commands

F(d) Position a cylinder of length d along the

h-axis, the turtle is translated to the

F(d,r) Position a cylinder with of length d

and radius r

-



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8. L-Systems

8.19

The 3D-Turtle Commands

+() Rotate around the u-axis by angle

&() Rotate around the r-axis by angle

o a e aroun e -ax s y ang e

| Turn around by 180, - the same as+(180)

[ ] Have the same effect as for the 2d-turtle


Turtle commands can be extended byarbitrary parameters

The 3D-Turtle

A 3D-extentionof the Hilbert

Cubes

inbetweenrepresentnodes



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8. L-Systems

8.20

A Tree Model

Monopodial branching structures

:A(1,10)p1:A(l,w) F(l,w)[&(a0)B(lr2,wwr)]

/(d)A(lr1,wwr)p2:B(l,w) F(l,w)[+(-a1)$C(lr2,wwr)]

C(lr1,wwr)p3:C(l,w) F(l,w)[+(a1)$B(lr2,wwr)]


B(lr1,wwr)The $ command turns the turtle back into ahorizontal position

A Tree Model

Parameters of the model:

r1: Contraction ratio for the trunk

2

a0: Branching angle from the trunk

a1: Branching angle for lateral axesd: Divergence angle


r


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8. L-Systems

8.21

A Tree Model


Development of Inflorescences

Correct simulation of development in discretesteps



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8. L-Systems

8.22

Development of Inflorescences


Phyllotaxis

Patterns of arrangements (leaves, seeds, ...)

Connected to Fibonacci numbers



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8. L-Systems

8.23

Phyllotaxis


Environment Sensitive Growth

Greensvoxel

automata

Not possiblewithpL-systems



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8. L-Systems

8.24

Environmentally-Sensitive L-Systems

The propagation of signals is used to prunebranches against a surrounding surface



Query modules check the environment

They are evaluated after each derivation

turtles state

These values can be used to selectproductions (e.g. trigger a signal,terminate growth)



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8. L-Systems

8.25


:A; p1:A F(1)?P(x,y)-A; p2:F(k) F(k+1)F(1)?P(*,*)-A

-,

F(2)?P(*,*)-F(1)?P(*,*)-A

F(2)?P(0,2)-F(1)?P(1,2)-A

F(3)?P(*,*)-F(1)?P(*,*)-F(1)?P(*,*)-A

F(3)?P(0,3)-F(1)?P(2,3)-F(1)?P(2,2)-A


(0,1)

(0,2) (1,2)

(0,3) (2,3)

(2,2)

Synthetic Topiary

Pruning of a 2D branching structure against asquare

// continue growth

p1

: A>?P(x,y):!prune(x,y) @F/(180)A// prune last segment

p2: A>?P(x,y): prune(x,y) T%p3: F>T S


// signal back propagation

p4: F>S SFp5: S // delete signalp6: @>S [+FA?P(x,y)] // triggers growth


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8. L-Systems

8.26

Synthetic Topiary


Synthetic Topiary



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8. L-Systems

8.27

Synthetic Topiary


Open L-Systems

Extension to general querymodules and auxilary data

Check for self intersection

Realized Greens voxelspace automata

Check for biolo ical


relevant conditions (soilcomposition, water, light,wind, ... )


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8. L-Systems

8.28

Open L-Systems


Fractals vs. Graftals

L-systems are a more general modelingtechnique for recursive defined objects,

Plants are not fractals, because:

They do not have infinite detailThey only have a very limited degree of selfsimilarity


They have not the scaling properties offractals


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8. L-Systems

8.29

Fractals vs. Graftals

PL-Systems are suitable for modelingobjects with a repetitive structure:

Terrains (fractal landscapes)

Linear fractals (IFS-attractors)

Architecture

Seashells


Particle systems

Efficient Rendering of pL-Systems

Use Directed Cyclic Graphs (DCG) as objectrepresentation

whole scene in memory

Similar to Hart and deFantis method for raytracing of linear fractals and Kajiyas methodfor fractal terrain


Problem: L-Systems are usually notcontractive


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8. L-Systems

8.30


Components

Selection node: Each symbol is,

joins all the productions of the symboland selects the proper one in each cycle

Transformation node: Map into localcoordinate system


Calculation node: Modify parameters,which affect production selection andtransformations


Optimization not trivial because of non-contractive mappings

equivalent with those of Hart and deFanti

Accumulate bounding boxes fortransformation nodes in pre-processing anduse their union for each traversal depth


Fill a grid during pre-processing

Grid resolution is used to balance memoryconsumption and rendering performance


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8. L-Systems

8.31


PL-System for Sierpinski tetrahedron

{c=6} S if(c>0,2,1)

2: S {c=c-1}(move(0.5,0.5,0.5)

uscale(0.5)S move(-0.5,-0.5,0.5)

uscale(0.5)S - -


. , . , .

uscale(0.5)S move(-0.5,0.5,-0.5)

uscale(0.5)S)


DCG for Sierpinski tetrahedron

c = 6

S

c > 0 c


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8. L-Systems

8.32


PL-System for a simple branching structure

{c=6} TR if(c>0,2,1)

2: TR {c=c-1}

(uscale(0.9) cylinder move(0,0,0.9)rotz(27)

(rotx(26)TR rotx(-32)TR))



DCG for a simple branchingstructure

TR

c > 0 c


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8. L-Systems

8.33


Ray Tracing:

Traverse graph for each ray

follow corresponding path

Evaluate calculation nodes

Apply inverse transformations oftransformation nodes to ray and accumulate


themIntersect ray with primitive object

Back transform normal vector




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8. L-Systems

Web Sites

www.cg.tuwien.ac.at/research/rendering/csg-graphs/index.html

.

www.xfrogdownloads.com


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Fractals -8- Dr Christoph Traxler.pdf