Lecture 37 of 42

Computing & Information SciencesKansas State University

Lecture 37 of 42CIS 636/736: (Introduction to) Computer Graphics

Lecture 37 of 42

Monday, 28 April 2008

William H. Hsu

Department of Computing and Information Sciences, KSU

KSOL course pages: http://snipurl.com/1y5gc

Course web site: http://www.kddresearch.org/Courses/CIS636

Instructor home page: http://www.cis.ksu.edu/~bhsu

Readings:

Handouts from Barnsley

Intro to Fractals and L-Systems



Computer Graphics 11:3D Object Representations –

Octrees & Fractals

Computer Graphics 11:3D Object Representations –

Octrees & Fractals

Course Website: http://www.comp.dit.ie/bmacnamee



FractalsFractals

All of the modelling techniques covered so far use Euclidean geometry methods Objects were described using equations

This is fine for manufactured objects But what about natural objects that have irregular or fragmented

features? Mountains, clouds, coral…

“Clouds are not spheres, mountains are not cones, coastlines are not circles and bark is not smooth, nor does lightning travel in a straight line.”

Benoit Mandelbrot



Fractal Geometry Methods & Procedural Modelling

Fractal Geometry Methods & Procedural Modelling

Natural objects can be realistically described using fractal geometry methods

Fractal methods use procedures rather than equations to model objects - procedural modelling

The major characteristic of any procedural model is that the model is not based on data, but rather on the implementation of a procedure following a particular set of rules

Modelling On The Fly!



FractalsFractals

A fractal object has two basic characteristics: Infinite detail at every point A certain self similarity between object parts and the overall features

of the object

The Koch CurveMandelbrot Set Video From:

http://www.fractal-animation.net/ufvp.htm



Generating FractalsGenerating Fractals

A fractal object is generated by repeatedly applying a specified transform function to points in a region of space

If P0 = (x0, y0, z0) is a selected initial position, each iteration of a transformation function F generates successive levels of detail with the calculations:

In general the transformation is applied to a specified point set, or to a set of primitives (e.g. lines, curves, surfaces)

... , , , 231201 PFPPFPPFP



Generating Fractals (cont…)Generating Fractals (cont…)

Although fractal objects, by definition have infinite detail, we only apply the transformation a finite number of times

Obviously objects we display have finite dimension – they fit on a page or a screen

A procedural representation approaches a true representation as we increase the number of iterations

The amount of detail is limited by the resolution of the display device, but we can always zoom in for further detail



Example: The Koch SnowflakeExample: The Koch Snowflake

Imag

es t

aken

fro

m H

earn

& B

aker

, “C

ompu

ter

Gra

phic

s w

ith O

penG

L” (

2004

)



Example: FernsExample: Ferns

Very similar techniques can be used to generate vegetation



Fractal DimensionFractal Dimension

The amount of variation in the structure of a fractal object is

described as the fractal dimension, D More jagged looking objects have larger fractal dimensions

Calculating the fractal dimension can be difficult, especially for particularly complex fractals

We won’t look at the details of these calculations



Types Of FractalsTypes Of Fractals

Fractals can be classified into three groups Self similar fractals

These have parts that are scaled down versions of the entire objectCommonly used to model trees, shrubs etc

Self affine fractalsHave parts that are formed with different scaling parameters in each

dimensionTypically used for terrain, water and clouds

Invariant fractal setsFractals formed with non-linear transformationsMandelbrot set, Julia set – generally not so useful



Random Midpoint Displacement Methods For Topography

Random Midpoint Displacement Methods For Topography

One of the most successful uses of fractal techniques in graphics is the generation of landscapes

One efficient method for doing this is random midpoint displacement



Random Midpoint Displacement Methods For Topography (cont…)

Random Midpoint Displacement Methods For Topography (cont…)

Easy to do in two dimensions Easily expanded to three dimensions to generate terrain

Can introduce a roughness factor H to control terrain appearance Control surfaces can be used to start with a general terrain shape

Terrain generation demo:http://world.std.com/~bgw/applets/1.02/MtFractal/MtFractal.html



Fractals In Film Special EffectsFractals In Film Special Effects



CS G140Graduate Computer Graphics

CS G140Graduate Computer Graphics

Prof. Harriet FellSpring 2007

Lecture 9 – March 26, 2007



FractalsFractals

The term fractal was coined in 1975 by Benoît Mandelbrot, from the Latin fractus, meaning "broken" or "fractured".

(colloquial) a shape that is recursively constructed or self-similar, that is, a shape that appears similar at all scales of magnification.

(mathematics) a geometric object that has a Hausdorff dimension greater than its topological dimension.



Mandelbrot SetMandelbrot Set

Mandelbrotset, rendered with Evercat's program.



Mandelbrot SetMandelbrot Set



What is the Mandelbrot Set?What is the Mandelbrot Set?

2

:cf

z z c

0

where is the -fold composition of with itself.

nc

nc c

f

f n f

We start with a quadratic function on the complex numbers.

The Mandelbrot Set is the set of complex c such that



ExampleExample

2

2

2

2 2

1

0 1 0 1 1 1 0

1 odd0

0 even

2 4 1 3 2 3 9 1 8

2 tend to as tends to .

1 2 2 4 1 3

tend to as tends to .

n

n

n

f z z

f f f

nf

n

f f f

f n

f i i f i f

f i n



(Filled-in) Julia Sets(Filled-in) Julia Sets

2

:cf

z z c

The Julia Set of fc is the set of points with 'chaotic' behavior under iteration.The filled-in Julia set (or Prisoner Set), is the set of all z whos orbits do not tend towards infinity. The "normal" Julia set is the boundary of the filled-in Julia set.

c = –1 c = –.5 +.5i c = – 5 +.5i



Julia Sets and the Mandelbrot SetJulia Sets and the Mandelbrot Set

Some Julia sets are connected others are not.

The Mandelbrot set is the set of c for which the Julia set of fc(z) = z2 + c is connected.

Map of 121 Julia sets in position over the Mandelbrot set (wikipedia)



A fractal is formed when pulling apart two glue-covered acrylic sheets.A fractal is formed when pulling apart two glue-covered acrylic sheets.



Fractal Form of a Romanesco Broccoli

photo by Jon Sullivan Fractal Form of a Romanesco Broccoli

photo by Jon Sullivan



Time for a BreakTime for a Break



L-SystemsL-Systems

An L-system or Lindenmayer system, after Aristid Lindenmayer (1925–1989), is a formal grammar (a set of rules and symbols) most famously used to model the growth processes of plant development, though able to model the morphology of a variety of organisms.

L-systems can also be used to generate self-similar fractals such as iterated function systems.



L-System ReferencesL-System References

Przemyslaw Prusinkiewicz & Aristid Lindenmayer, “The Algorithmic Beauty of Plants,” Springer, 1996.

http://en.wikipedia.org/wiki/L-System



L-System GrammarL-System Grammar

G = {V, S, ω, P}, where V (the alphabet) is a set of variables S is a set of constant symbols ω (start, axiom or initiator) is a string of symbols from V defining the initial

state of the system P is a set of rules or productions defining the way variables can be replaced

with combinations of constants and other variables. A production consists of two strings - the predecessor and the successor.



L-System ExamplesL-System Examples

Koch curve (from wikipedia) A variant which uses only right-angles.

variables : F constants : + − start : F rules : (F → F+F−F−F+F)

Here, F means "draw forward", + means "turn left 90°", and - means "turn right 90°" (see turtle graphics).



Turtle GraphicsTurtle Graphics

class Turtle { double angle; // direction of turtle motion in degrees double X; // current x position double Y; // current y position double step; // step size of turtle motion boolean pen; // true if the pen is down

public void forward(Graphics g) // moves turtle forward distance step in direction angle

public void turn(double ang)// sets angle = angle + ang;

public void penDown(), public void penUp()// set pen to true or false}



My L-System Data FilesMy L-System Data Files

Koch Triangle Form // title4 // number of levels to iterate90 // angle to turnF // starting shapeF:F+F-F-F+F // a rule

F F+F-F-F+F F+F-F-F+F+F+F-F-F+F-F+F-F-F+F-F+F-F-F+F+F+F-F-F+F

Go to Eclipse



More VariablesMore Variables

Dragon When drawing, treat L and R just like F.1090LL:L+R+R:-L-R

L L+R+ L+R+ + -L-R + L+R+ + -L-R + + - L+R+ - -L-R +



A Different AngleA Different Angle

Sierpinski Gasket660RL:R+L+RR:L-R-L

R L-R-L R+L+R- L-R-L -R+L+R



Moving with Pen UpMoving with Pen Up

Islands and Lakes290F+F+F+FF:F+f-FF+F+FF+Ff+FF-f+FF-F-FF-Ff-FFFf:ffffff // f means move forward with the pen up

F+F+F+F

next slide

F+f-FF+F+FF+Ff+FF-f+FF-F-FF-Ff-FFF



Islands and LakesOne Side of the BoxIslands and Lakes

One Side of the Box

F+f-FF+F+FF+Ff+FF-f+FF-F-FF-Ff-FFF



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

Using a Stack to Make TreesUsing a Stack to Make Trees

Tree1 [ push the turtle state onto the stack4 ] pop the turtle state from the stack22.5FF:FF-[-F+F+F]+[+F-F-F]

and I add leaves here



Stochastic L-Systemshttp://algorithmicbotany.org/lstudio/CPFGman.pdf

Stochastic L-Systemshttp://algorithmicbotany.org/lstudio/CPFGman.pdf

seed: 2454 // different seeds for different treesderivation length: 3axiom: FF--> F[+F]F[-F]F : 1/3F--> F[+F]F : 1/3F--> F[-F]F : 1/3



3D Turtle Rotations3D Turtle Rotations

Heading, Left, or, Up vector tell turtle direction.+(θ) Turn left by angle θ◦ around the U axis.−(θ) Turn right by angle θ◦ around the U axis.&(θ) Pitch down by angle θ◦ around the L axis.∧(θ) Pitch up by angle θ◦ around the L axis.\(θ) Rollleftbyangleθ◦ around the H axis./(θ) Roll right by angle θ◦ around the H axis.| Turn around 180◦ around the U axis. @v Roll the turtle around the H axis so that H and U lie in a common vertical plane with U closest to up.



A Mint http://algorithmicbotany.org/papers/

A Mint http://algorithmicbotany.org/papers/

A model of a member of the mint family that exhibits a basipetalflowering sequence.



Myles Akeem SingletonCentral Illinois Chapter

Myles Akeem SingletonCentral Illinois Chapter

National BDPA Technology Conference 2006Los-Angeles, CA

Examining the World of FractalsExamining the World of Fractals



Content of presentationContent of presentation

Introduction to fractals L-systems/Production rules Plant images Turtle geometry Conclusion



Introduction to fractalsIntroduction to fractals

Fractal Geometric Self-similar Has fractional dimension

Categorized under chaos science - fractal geometry 1975 - Benoît Mandelbrot defined the term fractal from the Latin

fractus, “broken” or “fractured”



Example of self-similarityExample of self-similarity



Koch Snowflake iterationsKoch Snowflake iterations



Julia set graphicJulia set graphic



Introduction to L-systemsIntroduction to L-systems

Fibonacci Thu-Morse Paperfolding Dragon curve Turtle graphics Branching Bracketed Several biological forms are branched, fragmented, or cellular in

appearance and growth Example where a trunk emerges from a branch:



Production rulesProduction rules

1968 - biologist Aristid Lindenmayer invents the L-system formula Used as a grammar to model the growth pattern of a type of

algae Set of production rules:

Rule 1: a → ab

Rule 2: b → a



Deterministic, context-free Lindenmayer system (D0L system)

Deterministic, context-free Lindenmayer system (D0L system)

Rule 1: a → ab

Rule 2: b → a

b → a a → ab ab → aba aba → abaab abaab → abaababa



Ben Hesper and Pauline HogewegBen Hesper and Pauline Hogeweg

Two of Lindenmayer’s graduate students Tested to see if L - systems could resemble botanic forms Images controlled by special characters would draw an image onto a screen

F → move forward one, drawingf → move forward one, without drawing+ → rotate clockwise by a given angle- → rotate counterclockwise by a given angle[ → push into stack] → pop from stack



Koch Island example“F → F + F - F - FF + F + F - F”

Koch Island example“F → F + F - F - FF + F + F - F”

F → move forward one, drawing

+ → rotate clockwise by a given angle

- → rotate counterclockwise by a given angle



Plant imagesPlant images

Adding a cursor stack system branching is gained Allows for the creation of plant-

like images

Mimics the structure of trees, bushes and ferns



Push/pop operations at workPush/pop operations at work

Angle 45

Axiom F

F = F [ + F ] F



Variables, constants, start words, and rulesVariables, constants, start words, and rules

Variables - symbols denoting replaceable elements Constants - symbols denoting fixed elements Start words - define how the system begins Rules - define how to replace variables with other variables or

constants



Turtle geometryTurtle geometry

Form of Logo programming Created 1967 at BBN, a Cambridge research firm, by Wally Feurzeig and

Seymour Papert

Grammar:nF - “n” steps forwardnB - “n” steps backaR - turn a degrees rightaL - turn a degrees leftConstants = {nF, nB, aR, aL, Stop}Variables = {, , , ...}Start = (none)



Turtle path exampleTurtle path example

<Path> → 5F 90R <Path><Path> → 5F 90R <Path><Path> → 5F 90R <Path><Path> → 5F 90R <Path><Path> → 5f <Path><Path> → 5F 90R <Path><Path> → 5F 90R <Path><Path> → 5F STOP

Production rules:F → move forward, drawingF → move forward, without drawingnF → “n” steps forwardnB → “n” steps backaR → turn “a” degrees rightaL → turn “a” degrees left <Path> denotes the part of the turtle's trail that is not specified Moves are represented by the transactions

Turtle graphic generated



ConclusionConclusion

Fractal uses Model many different objects and shapes Scientific modeling Creating graphic designs for clothes Multimedia 3-D artwork

Music pioneers of this research are learning how to apply the application of fractals to create new styles of music Uses a recursive process Algorithm is applied multiple times to process its previous output Provides very abstract musical results Becoming one of the most exciting fields of new music research

The limits of fractal will continue to stretch

Lecture 37 of 42

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