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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