A realization of a Poisson Point Process on $\mathbb{R}^2$ plane

Question

I am trying to visualize a 2 D homogenous PPP by plotting the realization in MATLAB.

I want to plot points over the $\mathbb{R}^2$ plane in MATLAB, a 2D Homogenous Poisson Point Process PPP is defined by its intensity

$$ \Phi_1 \text{ with intensity } \lambda_1 $$

For example lets take $$\lambda_1 = (80^2\pi)^{-1}$$

I appreciate any advice.

Edit based on answer below

   L1=10 ;
   L2=10;
intensity=(80^2*pi)^(-1);
parameter=L1*L2*intensity;
N=poissrnd(parameter,1,1);
x=unifrnd(1,10,1,N);
y=unifrnd(1,10,1,N)

Problem is that $N$ is zero ? Any advice?

Answer 1

To visualize this on a finite display, you will need to restrict to a rectangle $R$, say $[x_1, x_2] \times [y_1, y_2]$. The number of points $N$ in the PPP contained in $R$ is a Poisson random variable, where the mean is $\mu = \text{Area}(R)\cdot \lambda_1$. In MATLAB you can simulate a Poisson random variable using the poissrnd function. Once you have simulated a value for $N$, you then simply simulate $N$ independent random points in $R$, the x and y coordinates each having a uniform distribution.