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Turtle Graphics. Objectives To design and construct the class Turtle. To give some details about recursive methods. To present the principles of Turtle Graphics. To construct some Turtle images: trees, Koch curve, Serpinski curve,... . Working with Java classes.

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Turtle Graphics

Objectives

To design and construct the class Turtle.

To give some details about recursive methods.

To present the principles of Turtle Graphics.

To construct some Turtle images: trees, Koch curve, Serpinski curve,...

Computer Graphics - Lecture 6

Working with Java classes

Constructing a class /Constructing the object this.

- The class variables: private, …

- The class constructors:

- Same name as the class; do not have returning type.
- Initialise the class’ variables.
- The class methods: do an action on the class data.

- Type of methods: get/set, draw/print, ….
- Action on this non-static;
Create an object (call a constructor) :

NameClass obj = new NameClass(a1,a2,…);

Call a method

obj.nameMethod(a1,a2,…);

NameClass.nameMethod(a1,a2,…);

Computer Graphics - Lecture 6

Turtle-like Graphics Pens

What can turtles (as animals) do?

Move forward or backward.

Turn left or right.

Retreat the head into the shell.

Variables used for our Turtle-like Graphics Pens.

x, y give the turtle pen coordinates. Work with doubles

dir the direction to go in radians.

pen the pen’s status (pen=1 draws line)

Computer Graphics - Lecture 6

Design of the Class Turtle

public void moveTo(int x0,int y0)

{x=x0;y=y0;}

public void up(){pen=0;}

public void down(){pen=1;}

public void left(double a){dir+=a;}

public void right(double a){dir-=a;}

public void forward(int d,Graphics g){}

public void backward(int d,Graphics g)

{forward(-d,g);}

}

A Turtle object is a pen such that:

- Has a direction to go

- Can draw lines.

public class Turtle{

public final static double PI=Math.PI;

private double dir;

private int pen;

private double x,y;

public Turtle(int x0,int y0,int dir0);

public Turtle();

public double getDir(){return dir;}

public int getX(){return x};

public int getY(){return y};

Computer Graphics - Lecture 6

Implement and Test the Class Turtle

import java.awt.*;

import java.applet.*;

public class L6Appl1 extends Applet{

Turtle t = new Turtle();

final double PI=Math.PI;

public void paint(Graphics g){

t.moveTo(20,20);

t.forward(100,g);t.right(PI/2);

t.forward(100,g);t.right(PI/2);

t.forward(100,g);t.right(PI/2);

t.forward(100,g);t.right(PI/2);

}

}

See the execution

public Turtle(int x0,int y0,double dir0){

x=x0;y=y0;dir=dir0;

pen=1;

}

public Turtle(){

x=250;y=250;dir=0;pen=1;

}

public void forward(int d,Graphics g){

double x1,y1;

x1=(int)(x+d*Math.cos(dir));

y1=(int)(y+d*Math.sin(dir));

if (pen==1) g.drawLine((int)x,(int)y,(int)x1,(int)y1);

x=x1;y=y1;

}

Computer Graphics - Lecture 6

Recursive Methods

A method can call any (known) methods from the class or libraries.

A method is recursive when invokes itself.

Important Rule:

A recursive method must have a termination step that solves directly the problem.

type myRecMethod(type1 arg1, type2 arg2, …. ){

if (termination) {calculate directly the result res; return res; }

…

myRecMethod(a1,a2,…)

}

int sum (int n, int [] a){

if(n==0) return 0; // termination step

int s1 = sum(n-1,a); // recursive call

return s1+a[n-1];

}

Computer Graphics - Lecture 6

Principles of Turtle Geometry

1. Define Recursively the figure Fn.

- termination step: give the form of F0

- define Fn based on Fn-1.

2. Use a Turtle object to draw the figure.

- the direction of the Turtle object must be the same.

Turtle Geometry represents the simplest way to construct fractals.

Computer Graphics - Lecture 6

Binary Tree (1)

Consider T(n,l) the binary tree.

n - the order n; l - the length.

T(0,l)=a line with the length l.

T(n,l) is defined as follows:

- construct the trunk

- left 45 (PI/4)

- construct T(n-1,l/2)

- right 90 (PI/2)

- construct T(n-1,l/2)

- left 45 (PI/4)

- go back at the root

public void tree(int n, int l, Graphics g){

if(l==1 || n==0) return;

t.forward(l,g);

t.left(PI/4);tree(n-1,l/2,g);

t.right(PI/2);tree(n-1,l/2,g);

t.left(PI/4);

t.backward(l,g);

}

Computer Graphics - Lecture 6

import java.applet.*;

import Turtle;

public class L6Appl2 extends Applet{

TextField textField1, textField2;

Turtle t = new Turtle();

public void init(){

textField1 = new TextField(10);

textField2 = new TextField(10);

add(textField1); add(textField2);

textField1.setText("2");

textField2.setText("200");

}

Binary Tree (2)public void paint(Graphics g){

String s = textField1.getText();

int order = Integer.parseInt(s);

s = textField2.getText();

int length = Integer.parseInt(s);

t.moveTo(10,300);

tree(order,length,g);

}

public boolean action(Event event, Object arg){ repaint(); return true;

}

The declaration of the method Tree

}

Execution

Computer Graphics - Lecture 6

Consider K(n,l) the Koch’s curve.

K(0,l)= a line.

K(n,l) is defined as follows:

- construct K(n-1,l/3);

- left 60 (PI/3); construct K(n-1,l/3)

- right 120 (2PI/3); construct K(n-1,l/3)

- left 60 (PI/3); construct K(n-1,l/3)

The snow flake F(n,k)

- construct K(n,l); left 120 (2PI/3);

- construct K(n,l); left 120 (2PI/3);

- construct K(n,l); left 120 (2PI/3);

F(n,k) is a fractal representing an infinite curve bounding a finite area.

Koch’s Fractalpublic void koch(int n, int l, Graphics g){

if(l<2 || n==0){t.forward(l,g);return;}

koch(n-1,l/3,g);

t.left(PI/3);koch(n-1,l/3,g);

t.right(2*PI/3);koch(n-1,l/3,g);

t.left(PI/3);koch(n-1,l/3,g);

}

public void flake(int n, int l, Graphics g){

for(int i=0;i<3;i++){

koch(n,l,g); t.left(2*PI/3);

}

Computer Graphics - Lecture 6

Sierpinski’s Fractal (curve)

public void s(int n, int l, Graphics g){

int d=(int)l/Math.sqrt(2);

if(n==0)return;

s(n-1,l,g);t.right(PI/4);t.forward((int)d,g);

t.right(PI/4);s(n-1,l,g);

t.left(PI/2);t.forward(l,g);t.left(PI/2);

s(n-1,l,g);

t.right(PI/4);t.forward((int)d,g);t.right(PI/4);

s(n-1,l,g);

}

Consider S(n,l) the Sierpinski’s curve.

Find d=sqrt(l);

S(0,l) = nothing.

S(n,l) is defined as follows:

- construct S(n-1,l); right 45; forward d

- right 45; construct S(n-1,l);

- left 90; forward l; left 90;

- construct S(n-1,l);

- right 45;forward d; right 45

- construct S(n-1,l);

S(n,k) is a fractal representing an infinite curve bounded by finite area.

Computer Graphics - Lecture 6

Problems to Solve

Write an Applet to draw the Curve C(n,l).

C(0,l) is a line with the length l.

C(n,l) is defined

construct C(n-1,l); right 90;

construct C(n-1,l);left 90;

Define yourself and construct a tree with 4 branches.

Problems to read:

Recursive Methods: Java Black Book, Using recursion, pp 192.

An advanced topic about Turtle fractals:

http://www.math.sunysb.edu/~scott/Book331/turtle_in_fractal.html

Computer Graphics - Lecture 6

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