I have a set of points S(0) that is closer to (0,0) than the edges of of a square with vertices (1,1),(1,-​1),(-1,-1)​,(-1,1). I have the code and now i just need to modify it to estimate the area of S(0)

close all

rng('Shuffle'); 

NumberInside = 0; 

PiEstimate = zeros(500,1);

for k=1:500

x = -1+2*rand(100,1); 

y = -1+2*rand(100,1); 

     NumberInside = NumberInside + sum(x.^2 + y.^2 <= 1);

     PiEstimate(k) = (NumberInside/(k*100))*4;    
  end

plot(PiEstimate)

title(sprintf('Monte Carlo Estimate of Pi = %5.3f',PiEstimate(500)));

xlabel('Hundreds of Trials')

[KSSV]

Not clear......can you elaborate?

[Tony Phan]

Sorry. So i basically have a square board and inside of that i have a set of points S(0), which is similar to a circle but not quite a circle(think of a dartboard almost), I need help modifying the code to estimate the area of S(0)

[Jos (10584)]

What is your definition of the area of a set of points?

Answer 1

[Jyotish Robin]

Hi Tony!

From your query, it looks like the set of points S(0) is defined by the (x,y) coordinates that are randomly generated in your code.

But, does it look similar to a circle? No is my hunch. If you wanted to generate a set of points that look similar to a circle , you could use something similar to the below:

t = 2*pi*rand(n,1); % n=number of points 
r = R*sqrt(rand(n,1)); % R =radius of circle
x = x0 + r.*cos(t); %(x0,y0) : center of circle
y = y0 + r.*sin(t);

Now, once you have the cluster of points, you can generate a boundary and then estimate the area within. To do this, make use of the boundary function. ( https://www.mathworks.com/help/matlab/ref/boundary.html )

[k,v] = boundary(_) returns a scalar v, which is the area which boundary k encloses.

 [k, v] = boundary(x,y);
 plot(x(k),y(k));

Based on your existing code's structure, it seems that you are trying to estimate pi from this area calculation.

Here, the returned 'v' will be an estimate of pi/4. (If R=0.5)

Hence 4 times 'v' will be an estimate of pi.

Hope it helps!

Thanks,

Jyotish