CCSU  Spring, 1999

Project 2

Posting Date:  February 2, 1999
Due Date:      Friday, March 9, 1999

Relevant Demos:  Demo 12: Parabola.java, Demo 13: ParabolaApplet.java, Demo 14: 
ParabolaApplication.java

This project comes in 3 parts.  It covers the material covered in Chapter 3 on 
control structures (if else, while), plus the material covered in chapter 4 on 
class Random, plus the material covered in class on Applet and Application 
graphics, plus the material covered in class from Chapter 5 on control 
structures (more on if else, for).

This project and demos 12 through 14 also illustrate an important class of 
methods for scientific computing, Monte Carlo methods.  In a Monte Carlo 
method, one uses a random process (pseudo-random process actually) to simulate 
something in nature or mathematics, usually something quite complex.  Monte 
Carlo methods are named after the famous casino in Monte Carlo.

In this project, you are going to use a Monte Carlo method to approximate the 
area of a circle of radius 1.  Since A = pi*r*r, and r = 1 here, the computed 
approximation to the area A will be approximately equal to the mathematical 
constant pi.

Here is how the method works:  Throw 10,000 darts at random at a square of side 
2, with vertices (corners) at (-1, -1), (1, -1), (1, 1) and (-1, 1).  Note that 
the origin (0, 0) is at the center of the square, and note also that you could 
draw a circle of radius 1 centered at the origin and have it just touch the 
edges of the square (be inscribed in the square).  Count the number of darts 
which hit the circle, and at the end, compute the fraction of darts which hit 
the circle by dividing the total number of hits by 10,000.  Now the fraction f 
of darts hitting the circle is equal to

     areaOfTheCircle/AreaOfTheSquare,

except for statistical error.  Now areaOfTheCircle = pi, and areaOfTheSquare = 
4.  Thus the fraction f is equal to pi/4, except for statistical error.  
Multiply f by 4 to get an approximation to pi,

     pi = 4*f

approximately.  (Approximately because of the statistical error.)

Here are the three parts of the project:

1. Modify Demo 12 so you compute an approximation to pi by computing the area 
of a circle of radius 1.  You will use the condition

    x*x + y*y <= 1

to tell when a random point lies inside the circle.  Call your class PiFinder.

2. Modify Demo 13 and part 1 so your program runs as an applet.  Call your 
class PiFinderApplet.  Draw the circle inscribed in the square, and display 
graphically where each 100th random point hits.

Hint: Drawing the circle is much simpler than drawing the parabola in demo 13.
All you have to do is use g.drawOval.  Look at the Storm applet in chapter 4
of the text to see an example of circle drawing using g.drawOval.  But draw
the circle in paint, not in start.  Drawing in the Applet start method the
way the text does often doesn't work in the most recent versions of the JDK.

3.  Modify Demo 14 and part 2 so your program runs as a GUI application.  Call 
your class PiFinderApplication.

Notes:

(1) Your files MUST be named PiFinder.java, PiFinder.class, 
PiFinderApplet.java, PiFinderApplet.class, PiFinderApplication.java and 
PiFinderApplication.class, so I can easily find them.

(2)   Each of your programs must begin with a remark box as described in 
project 1.

Due:  Friday, March 19, 1999 AT THE BEGINNING OF CLASS.