Lab 2: Recipies - Part 2

Released: 11:59 PM Sunday, June 7th, 2020.

Due: Monday, June 8th, 2020 by the end of recitation.

This lab is the second part of Project 1 - Calculating Pi using a computer simulation. Today we will focus on using conditionals to decide if a random point falls inside or outside the circle.

This lab is Part 2 for the first project, read the project description again if you need to jog your memory.

Your assignment for this lab is to solve these exercises and show them working to your TA by the end of the recitation. You don’t have to submit anything for this lab. Only the final implementation of the project will be submitted! Part 3 of the lab will contain instructions on how to submit your project.

In this lab, you will take your Lab 1 implementation and expand it with some code that allows you to decide if the points you generate fall inside or outside of the circle and the rectangle.

Part 1

Let’s start with the maths. How can you tell if a point (x, y) belongs to a circle? Well, it’s easy, don’t you remember from high school? 😏 Ok, review time!

How do we define a circumference?

A circumference is a geometric curve where the distance of every point to its centre is the same, and it’s equal to the radius of the circumference. So, if you have a circumference of radius r centred in (0, 0) (like we have for our simulation) the equation of the circumference is:

\[\sqrt{x^2+y^2}=r\]

Or equivalently (after we square both sides of the equation):

\[x^2+y^2=r^2\]

Easy! Right?

In fact, the equation says that point (x, y) is part of the circumference if its distance to (0, 0) - calculated as \(\sqrt{x^2+y^2}\) - equals the radius of the circumference \(r\).

It is slightly trickier if the circle is not centred in (0, 0) that’s why I asked you to generate the points around the coordinate (0, 0). :)

How do we define a circle?

A circle is both the circumference and all points inside it.

So, if you have a circle centred in (0, 0) (like we have for our simulation) a point that belongs to that circle, must be at a distance from the centre that is no larger than the circumference:

\[x^2+y^2<=r^2\]

In other words wink, wink:

Given the point (x, y) and the circle centred in (0, 0) of radius r.
The point is inside the circle if and only if \(x^2+y^2<=r^2\).
Else, the point is outside of the circle.

How to calculate x to the power of 2 in Java?

Great question! Java has a lot of useful mathematical functions ready for us to use. One of the is the pow function! And if we want to do something like raising 2 to the power of 10, we simply need to write:

double power = Math.pow(2.0, 10.0);

Or square a number:

double r = 3.0;
double power = Math.pow(r, 2.0);

Conditionals

In order to make our program behave differently based on values we calculate, or based on user input, we need to use conditionals. One of the most widespread conditionals (all programming languages that I’ve used have them!) is the if-then-else statement. That we saw in lecture last week! In case you don’t remember, this is what it looks like:

boolean condition;
if(condition)
{
  System.out.println("The condition is true");
}
else
{
  System.out.println("The condition is false");
}
System.out.println("This always runs");

Your turn

To estimate the value of \(\pi\), we need to be able to generate multiple random points, count them, and do some calculations. So, towards that purpose, lets make some modifications to the code we created for the previous lab.

1. Start by creating 2 variables named total and count in your Lab 1 program. You will use these variables to keep track of how many points you have generated, and how many were inside the circle.

2. Modify your program to check if the point (xCoordinate, yCoordinate) is within a circle centred in (0, 0) and with radius size/2. Where size is the length of the square input by the user. Make your program output a message stating if the point is within the circle or not

Example output using the seed number 1:

What is the size of the rectangle to be used in the calculation of Pi?
3.2
The coordinate of the point is: (0.728203776105178, 0.5863151096315055)
(0.728203776105178, 0.5863151096315055) is inside the circle of radius 1.6

2. Modify your program to generate 10 points. Try copy-pasting the code you have to generate a random point and checking if it is inside the circle 9 times.

Warning: Copy-pasting is a powerful technique to quickly replicate code we wrote before, but be careful: With great power… come great mistakes. Usually:

Declaring variable multiple times
Copy-pasting code that we did not intend.
Copy-pasting code that needs small changes, that we forget!
(joke!) What you just did! ;)

Example output using the seed number 1:

What is the size of the rectangle to be used in the calculation of Pi?
3.2
The coordinate of the point is: (0.7388102102505307, -0.28774140322495473)
(0.7388102102505307, -0.28774140322495473) is inside the circle of radius 1.6
The coordinate of the point is: (-0.9353125078089054, -0.5353054209295642)
(-0.9353125078089054, -0.5353054209295642) is inside the circle of radius 1.6
The coordinate of the point is: (1.4968189101571863, -1.580425016749564)
(1.4968189101571863, -1.580425016749564) is outside the circle of radius 1.6
The coordinate of the point is: (1.4838553504742646, 1.4075692441021115)
(1.4838553504742646, 1.4075692441021115) is outside the circle of radius 1.6
The coordinate of the point is: (1.4310237365222205, 1.398662876467103)
(1.4310237365222205, 1.398662876467103) is outside the circle of radius 1.6
The coordinate of the point is: (-0.3290421050089421, -0.4879423065500472)
(-0.3290421050089421, -0.4879423065500472) is inside the circle of radius 1.6
The coordinate of the point is: (-0.6590174975870824, 0.02074760744395228)
(-0.6590174975870824, 0.02074760744395228) is inside the circle of radius 1.6
The coordinate of the point is: (-1.2289053182954952, 0.8657148162533688)
(-1.2289053182954952, 0.8657148162533688) is inside the circle of radius 1.6
The coordinate of the point is: (0.5116566678189439, -1.0984099501764921)
(0.5116566678189439, -1.0984099501764921) is inside the circle of radius 1.6
The coordinate of the point is: (0.5116566678189439, -1.0984099501764921)
(0.5116566678189439, -1.0984099501764921) is inside the circle of radius 1.6

Think about it: Do you think it is a good idea to copy-paste your code like this? Can you think of any cons of doing this? For example:

What happens if you made a mistake? Now you have to change all 10 copies… And you know you’ll forget 1! :)
How many more space does your program take? In a small program like this it’s not a huge deal, but what about when you write a really big program?
What if you want the user to decide how many points should be generated?
What if the lab asks you for a change that must be made in every single copy????

About that…

3. We need to keep track of how many points fall into the circle and the total number of points…
Yikes!
Modify your code such that it adds 1 to variable total for each point it generates, and that it adds 1 to the variable count if the point is inside the circle.

4. Finally, print how many points fell inside the circle, the total of points you generated, and your estimate of \(\pi\).

Example output using the seed number 1:

What is the size of the rectangle to be used in the calculation of Pi?
3.2
The coordinate of the point is: (0.7388102102505307, -0.28774140322495473)
(0.7388102102505307, -0.28774140322495473) is inside the circle of radius 1.6
The coordinate of the point is: (-0.9353125078089054, -0.5353054209295642)
(-0.9353125078089054, -0.5353054209295642) is inside the circle of radius 1.6
The coordinate of the point is: (1.4968189101571863, -1.580425016749564)
(1.4968189101571863, -1.580425016749564) is outside the circle of radius 1.6
The coordinate of the point is: (1.4838553504742646, 1.4075692441021115)
(1.4838553504742646, 1.4075692441021115) is outside the circle of radius 1.6
The coordinate of the point is: (1.4310237365222205, 1.398662876467103)
(1.4310237365222205, 1.398662876467103) is outside the circle of radius 1.6
The coordinate of the point is: (-0.3290421050089421, -0.4879423065500472)
(-0.3290421050089421, -0.4879423065500472) is inside the circle of radius 1.6
The coordinate of the point is: (-0.6590174975870824, 0.02074760744395228)
(-0.6590174975870824, 0.02074760744395228) is inside the circle of radius 1.6
The coordinate of the point is: (-1.2289053182954952, 0.8657148162533688)
(-1.2289053182954952, 0.8657148162533688) is inside the circle of radius 1.6
The coordinate of the point is: (0.5116566678189439, -1.0984099501764921)
(0.5116566678189439, -1.0984099501764921) is inside the circle of radius 1.6
The coordinate of the point is: (0.5116566678189439, -1.0984099501764921)
(0.5116566678189439, -1.0984099501764921) is inside the circle of radius 1.6
7.0 points out of 10.0 are inside the circle.
The pi estimate is 2.8

Remember: The estimate of \(\pi\) is proportional to the ratio of number of points that fell inside/outside the circle. Read the project description for an in-depth explanation.

\[\pi = 4\times\frac{count}{total}\]

Good news: There are better ways of repeating your code several times! Soon…

Demo

Once you have finished, show the lab instructor that you have Lab 1 working.