To generate the locations (x, y) of 100 nodes within a radius of 1 kilometer using a Homogeneous Poisson Point Process (HPPP) with a given spatial density, you can follow these steps:
- Calculate the area of the circle: The area (A) of a circle with radius (r) is given by (A = \pi r^2).
- Determine the spatial density: The spatial density (\lambda) is the number of points per unit area. Given the number of nodes (N) and the area (A), you can calculate (\lambda) as (\lambda = \frac{N}{A}).
- Generate the points: Use the spatial density (\lambda) to generate the points within the circle.
Here is a MATLAB script to achieve this:
% Parameters
radius = 1; % Radius in kilometers
numNodes = 100; % Number of nodes
area = pi * radius^2; % Area of the circle
lambda = numNodes / area; % Spatial density
% Generate points using the Homogeneous Poisson Point Process
theta = 2 * pi * rand(numNodes, 1); % Random angles
r = radius * sqrt(rand(numNodes, 1)); % Random radii
% Convert polar coordinates to Cartesian coordinates
x = r .* cos(theta);
y = r .* sin(theta);
% Plot the points
figure;
scatter(x, y, 'filled');
hold on;
viscircles([0, 0], radius, 'LineStyle', '--'); % Draw the circle boundary
axis equal;
title('HPPP Generated Nodes within 1 km Radius');
xlabel('X (km)');
ylabel('Y (km)');
grid on;
Hope this helps!
