# Fractal by dffhrtcv3

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```									      Fractal

Han Jinshu
Department of Computer
Contents

 Why we introduce fractal?
 What is Fractal?
The properties of fractal are
what?
 How to draw fractal pictures?
Why we introduce fractal?
 Procedural Methods
 Reason 1: we need
realistic graphics.
•    Euclidean-geometry
•"Clouds are not
methods;                      spheres, mountains are
•   Surface;                      not cones, coastlines
•    Convex planar polygons;      are not circles, and
•    Euclidean metric space;
•    Manufactured objects:
bark is not smooth, nor
those that have smooth        does lightning travel in
surfaces and regular          a straight line."（B.B.
shapes.
•   Natural objects : irregular   Mandelbrot）
shapes or
fragmentedfeatures
Why we introduce fractal?
 Reason 2: rendering speed
• 10 million polygons per second; greater
speed; David statue
 Reason 3: database sizes
• more space; parameters;
 In response to these problems,
researchers had developed procedural
methods, which describe objects in an
algorithmic manner.
 美国斯坦福大学计算机系的著名图形
学专家Marc Levoy曾经带领他的30
人工作小组（包括美国斯坦福大学及
美国华盛顿大学的教师和学生）
 于1998～1999学年在意大利，专门
对文艺复兴时代的雕刻大师米开朗基
罗的众多艺术品进行扫描，保存其形
状和面片信息
 专门设计了一套硬件和软件系统
 数据量惊人，光David statue就有2
billion个多边形和7000张彩色图象，
总共需要72G的磁盘容量
Why we introduce fractal?
 Two of many possible approaches to
procedural modeling:
• Particle system
• Fractal geometry:
Fractal is a new branch of mathematics and
art. It approximate object with a few rules.
A new viewpoint, not only in computer
graphics, but also for different domain of
science. It’s one of the growth points of
nolinear science.
What is Fractal?
 Definition:
• B.B. Mandelbrot
A rough or fragmented geometric shape that
can be subdivided in parts, each of which is
(at least approximately) a reduced size copy
of the whole.
• Mathematical aspect
A set of points whose fractal dimension
exceeds its topological dimension.
• We use the results to draw fractal pictures.
What is Fractal?
 Properties of Fractal: Self-similarity
• Example 1: fern leaf

every little leaf part
of the bigger one has
the same shape as
the whole fern leaf .
each of little leaf
part is a reduced size
copy of the whole
What is Fractal?
Contractive Affine
• Example 2: Sierpinski Triangle

rule1 : x  0.5 x; y  0.5 y
rule2 : x  0.5 x  0.5; y  0.5 y
rule3 : x  0.5 x  0.25; y  0.5 y  0.5
What is Fractal?
• Example 3: coastline problem
• How long is the coast of Britain?
1)Long before the invention of computers,
2)British cartographers ,
3)Measure the length of British coast,
4)The coastline measured on a large scale
map was approximately half the length of
coastline measured on a detailed map. The
closer they looked, the more detailed and
longer the coastline became.
What is Fractal?
• Example 3: coastline problem
• Answer: Koch curve simulate coastline
What is Fractal?
• Example 3: coastline problem
• Answer: Koch curve simulate coastline
• Why the length is uncertain?
测量时所用的尺度不同. Koch曲线是一条无限长的线
折叠而成的。
 Key words:
Self-similarity;
Contractive affine;
Simple rulers <-> complex phenomena
What is Fractal?
 A novel idea:
simple rulers <-> complex phenomena
• database sizes:
Don’t need save many polygon parameters
• rendering speed:
Don’t render so many polygon
• realistic graphics:
generate beautiful fractal image to simulate
natural objects.
How to draw fractal pictures?
 A lot of different types of fractal.
• Iteration function system(IFS) (example 1+2)
• L-system (example 3)
How to draw fractal pictures?
 A lot of different types of fractal.
• Iteration function system(IFS) (example 1+2)
• L-system (example 3)
• Chaotic Systems
 Julia sets
 Mandelbrot set
How to draw fractal pictures?
 L-systems are a mathematical formalism
proposed by the biologist Aristid Lindenmayer
in 1968 as a foundation for an axiomatic theory
of biological development. More recently, L-
systems have found several applications in
computer graphics.
• Smith 1984; Prusinkiewicz and Hanan 1989;
Prusinkiewicz and Lindenmayer 1991
 Two principal areas include generation of
fractals and realistic modelling of plants.
How to draw fractal pictures?
 Central to L-systems, is the notion of
rewriting, where the basic idea is to
define complex objects by successively
replacing parts of a simple object using a
set of rewriting rules or productions.
 The rewriting can be carried out
recursively.
How to draw fractal pictures?
 Two steps
• Bulid string
• Fractals and graphic interpretation of strings
 Bulid string
• Initial string(axiom): F
 F: Move forward a step of length d. The
state of the turtle changes to (x',y',a),
where
x'= x + d cos(a) , y'= y + d sin(a).
A line segment between points (x,y) and
(x',y') is drawn.
 +: Turn left by angle b. The next state
of the turtle is (x,y,a+b).
 - : Turn left by angle b. The next state of
the turtle is (x, y,a-b)
How to draw fractal pictures?
 Bulid string -> A long string
• Initial string(axiom): F
• Rewriting rulers: F=F+F--F+F
• After on interation the following string would
result
F+F--F+F+ F+F--F+F-- F+F--F+F+ F+F--F+F
• Iteration time: n=3
How to draw fractal pictures?
 Bulid string -> A long string
 Fractals and graphic interpretation of strings
•   Turning angle: agd=2*PI/6
•   Length of line: d
•   Initial position
•   Show example(koch curves): koch.exe
How to draw fractal pictures?
 L-system is an effective method. We can
modify the parameters, in order to view the
difference.(5.3)
• Koch5, peanol, P2
 Recent usage of L-Systems is for the creation
of realistic looking objects that occur in
nature and in particular the branching
structure of plants.
How to draw fractal pictures?
 Plants have branches, how to describe them?
 [: Pop a state from the stack and make
it the current state of the turtle.
 ]: Push the current state of the turtle onto a
pushdown stack.
 Show example
 More rulers

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