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Fractal

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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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posted:8/19/2012
language:English
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