1 What did we learn before?

2 line and segment generation

3 Filled region

4 Curves and surfaces

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7 Geometric Transformations

8 clipping

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10 3D modeling

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12 Regular objects ’ representation: Regular objects ’ representation: Euclidean-geometry methods. Euclidean-geometry methods. Irregular objects ’ representation: Irregular objects ’ representation: Fractal-geometry methods. Fractal-geometry methods. Review :

13 Chapter 8 Fractal Geometry 分形几何

what are fractals some pictures and animation films some pictures and animation films

15 Definitions of fractals Definitions of fractals 1. B.B.Mandelbrot (In 1982) 1. B.B.Mandelbrot (In 1982) A fractal is by definition a set for which the Hausdorff- Besicovitch dimension strictly exceeds the topological dimension. 强调维数不是整数，是分数，又称分数维 强调维数不是整数，是分数，又称分数维

16 Koch curve Koch curve similarity dimension is 1.26 similarity dimension is 1.26

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19 middle third Cantor set middle third Cantor set similarity dimension: 0.68 similarity dimension: 0.68

20 Sierpinski triangle Sierpinski triangle similarity dimension : 1.58 similarity dimension : 1.58

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22 Definitions of fractals Definitions of fractals 1. B.B.Mandelbrot (In 1982) 1. B.B.Mandelbrot (In 1982) A fractal is by definition a set for which the Hausdorff- Besicovitch dimension strictly exceeds the topological dimension. 强调维数不是整数，是分数，又称分数维 强调维数不是整数，是分数，又称分数维 2. B.B.Mandelbrot (In 1986) 2. B.B.Mandelbrot (In 1986) A fractal is shape made of parts similar to the whole in some way. A fractal is shape made of parts similar to the whole in some way. 强调局部与整体自相似性 强调局部与整体自相似性

23 peano n=1n=2 n=3 n=4

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Fractal Properties F has a fine structure, ie detail on arbitrarily small scales. F has a fine structure, ie detail on arbitrarily small scales. F has too irregular to be described in traditional geometrical language, both locally and globally. F has too irregular to be described in traditional geometrical language, both locally and globally. Often F has some form of self-similarity, perhaps approximate or statistical. Often F has some form of self-similarity, perhaps approximate or statistical. Usually, the fractal dimension of F is greater than its topological dimension. Usually, the fractal dimension of F is greater than its topological dimension. In most cases of interest of F is defined in a very simple way, perhaps recursively. （递归迭代） In most cases of interest of F is defined in a very simple way, perhaps recursively. （递归迭代）

Fractal Dimension

27 Fractal similarity dimension: Fractal similarity dimension: ⑴ the straight-line segment scale number length scale number length (r) (N) (r) (N) 1/ / / /3 3 1 … … … … … … 1/n n 1 1/n n 1 1=N · r 1 1=N · r 1

28 ⑵ square (s=1) scale number area scale number area (r) (N) (s) (r) (N) (s) 1/ / / /3 9 1 … … … … … … 1/n n 2 1 1/n n 2 1 1=N · r 2 1=N · r 2

29 ⑶ a cube (v=1) scale number volume scale number volume (r) (N) (s) (r) (N) (s) 1/ / / / … … … … … … 1/n n 3 1 1/n n 3 1 1=N · r 3 1=N · r 3

30 r — scaling factor N — the number of subparts N · r D =1 D= ㏒ N/ ㏒ (1/r) N · r D =1 D= ㏒ N/ ㏒ (1/r)

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32 · initiator — start with a given geometric shape 8.4 Geometric Construction of Deterministic Self-Similar Fractals · generator — subparts of the initiator are replaced with a pattern with a pattern

33 Basic idea: construction of the von koch each segment in (Fig.1) is replaced by an exact copy of the entire figure, shrunk by a factor of 3. The same process is applied to the segments in (Fig.2) to generate those in (Fig.3).

34 ① ② ③ ④ (x s, y s ) Angle :>0 counterclockwise direction <0 clockwise direction 60 0

35 global variables: int th; current value of the angle float x, y; x, y coordinates float d; the length of each segment d=L/m n m: 等分数 d=L/m n m: 等分数 n: iteration times n: iteration times

36 Void Generate – koch (n) //n : recursive depth { if (n=0) { x+=d*cos (th* /180) { x+=d*cos (th* /180) y+=d*sin (th* /180) y+=d*sin (th* /180) line to (x, y) ; line to (x, y) ; return ; return ; } Generate – koch (n-1); Generate – koch (n-1); th+=60 ; th+=60 ; Generate – koch (n-1); Generate – koch (n-1); th-=120 ; th-=120 ; Generate – koch (n-1); Generate – koch (n-1); th+=60 ; th+=60 ; Generate – koch (n-1); Generate – koch (n-1);}

37 n=0 d=L th=0 x=0 y=0 Generate – koch (0) Generate – koch (0) x=d, y=0 x=d, y=0 (0,0)(L,0)

n=1 Generate-koch(1) Generate-koch(0) th+=60; Generate-koch(0) th-=120 0 Generate-koch(0) th+=60 Generate-koch(0)

39 n=2 Generate-koch(2) n=1 Generate-koch(1) n=0 Generate-koch(0) th=0 x=0 y=0 d=L/3 2 x=0+dcosth=d y=0+dsinth=0 line to th+=60 Generate-koch(0) th=60 x=d y=0 x=d+dcosth y=0+dsinth line to th-=120 0 Generate-koch(0) th=-60 0 x=d+dcos60 0 y=dsin60 0 x=x+dcosth y=y+dsinth line to th+=60 0 Generate-koch(0) th=0 x=x+d y=y+0 line to … … …

40 th+=60 0 Generate-koch(1) n=0 Generate-koch(0) th=60 0 x=x+dcosth y=y+dsinth line to th+=60 0 Generate-koch(0) th=120 0 x=x+dcosth y=y+dsinth line to th-=120 0 Generate-koch(0) th=0 x=x+d y=y+0 line to th+=60 0 Generate-koch(0) th=60 0 x=x+dcosth y=y+dsinth line to … … … n=2 Generate-koch(2)

41 th-=120 0 Generate-koch(1) n=0 Generate-koch(0) th+=60 0 Generate-koch(0) th=0 x=x+d y=y+0 line to th+=120 0 Generate-koch(0) th= x=x+dcosth y=y+dsinth line to th+=60 0 Generate-koch(0) th=-60 0 x=x+dcosth y=y+dsinth line to th=-60 0 x=x+dcosth y=y+dsinth line to … … … n=2 Generate-koch(2)

42 Generate-koch(1) n=0 Generate-koch(0) th+=60 0 Generate-koch(0) th=60 0 x=x+dcosth y=y+dsinth line to th-=120 0 Generate-koch(0) th=-60 0 x=x+dcosth y=y+dsinth line to th+=60 0 Generate-koch(0) th=0 x=x+d y=y+0 line to th=0 x=x+d y=y+0 line to th+=60 0 n=2 Generate-koch(2)

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45 other kinds of Koch ① ② ③ ④ ⑤ ⑥ ⑦⑧ ⑨ D= ㏒ N/ ㏒ (1/r)= ㏒ 9/ ㏒ 3=2 ① ② ③ ④ ⑤ ⑥ ⑦ ⑧ D= ㏒ 8/ ㏒ 4=1.5

46 peano n=1n=2 n=3 n=4

47 6 Questions Map plotting based on fractal curves Map plotting based on fractal curves

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50 种植果树的山坡（韩云萍）

51 (a) (b) 果实和果树的构造（韩云萍）

年，美国《科学》杂志提出一个问题：英 国海岸线有多长？ 1967 年，美国《科学》杂志提出一个问题：英 国海岸线有多长？ Mandelbrot 对此问题的回答是：海岸线长度可以认为 是不确定的。 Mandelbrot 对此问题的回答是：海岸线长度可以认为 是不确定的。 对此问题的分析 : 对此问题的分析 : 如从高空飞行的飞机往下测量，测得的海岸线长度 为 x1 。当从低空飞行的飞机测得的海岸线长度为 x2 ， … ，越飞越低，测量的精度越来越高，测量值显 然有以下关系： X1<x2<x3< … 如从高空飞行的飞机往下测量，测得的海岸线长度 为 x1 。当从低空飞行的飞机测得的海岸线长度为 x2 ， … ，越飞越低，测量的精度越来越高，测量值显 然有以下关系： X1<x2<x3< … 如果让一个小虫沿海岸爬行，那末它所经过的曲折 更多，如果用分子、原子来测量，显然测得的 Xn 是天 文数字。这说明当对研究对象的观察越贴近，越仔细， 那么发现的细节就越多． 如果让一个小虫沿海岸爬行，那末它所经过的曲折 更多，如果用分子、原子来测量，显然测得的 Xn 是天 文数字。这说明当对研究对象的观察越贴近，越仔细， 那么发现的细节就越多．

53 但是在不同高度观察到的海岸线的曲折、复杂程 度又十分相近，也就是说，海岸线有自相似性。 Mandelbrot 用简单的 Koch 曲线来模拟英国海岸线 比用折线段来逼近海岸线要精确得多。

54 Koch 曲线的构造方法： 定义一个源多边形，称为初 始元（ initiator ），例如一 个直线段；再定义一个生成 多边形，称为生成元（ generator ）. 通过几何结构 的迭代，得到的极限曲线就 是一条 “ 处处连续处处不可 微的曲线 ” ．分析一下这条 极限曲线的长度，设直线长 度 L 为 1 ，有以下结果： 尺度 段数 长度 1/3 4 4/3 1/9 4 2 (4/3) 2 …… 1/3 n 4 n (4/3) n …

55 当 n  ∞ 时，长度（ 4/3 ） n  ∞ ，是一个不确 定值，这就是对 “ 英国海岸线有多长？ ” 的一个精 辟的回答。 当 n  ∞ 时，长度（ 4/3 ） n  ∞ ，是一个不确 定值，这就是对 “ 英国海岸线有多长？ ” 的一个精 辟的回答。

56 Measurement of length

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