1. Computer Graphics - Lecture 6
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,...
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Computer Graphics - Lecture 6

2. 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,…);
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Computer Graphics - Lecture 6

3. 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)
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Computer Graphics - Lecture 6

4. 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};
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Computer Graphics - Lecture 6

5. 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); }
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;
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Computer Graphics - Lecture 6

6. 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];
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Computer Graphics - Lecture 6

7. 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.
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Computer Graphics - Lecture 6

8. 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)
- 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); }
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Computer Graphics - Lecture 6

9. Computer Graphics - Lecture 6
Binary Tree (2)
import java.awt.*;
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"); }
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
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10. Computer Graphics - Lecture 6
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11. Computer Graphics - Lecture 6
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12. Computer Graphics - Lecture 6
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13. Computer Graphics - Lecture 6
Koch’s Fractal
public 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); }
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);
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)
The snow flake F(n,k)
- construct K(n,l); left 120 (2PI/3);
F(n,k) is a fractal representing an infinite curve bounding a finite area.
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14. Computer Graphics - Lecture 6
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15. Computer Graphics - Lecture 6
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16. Computer Graphics - Lecture 6
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17. Sierpinski’s Fractal (curve)
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
S(n,k) is a fractal representing an infinite curve bounded by finite area.
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); }
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18. Computer Graphics - Lecture 6
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19. Computer Graphics - Lecture 6
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20. Computer Graphics - Lecture 6
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21. 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:
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Computer Graphics - Lecture 6