Matlab, generate and plot a point cloud distributed within a triangle

Question

I'm trying to generate a cloud of 2D points (uniformly) distributed within a triangle. So far, I've achieved the following:

The code I've used is this:

N = 1000;
X = -10:0.1:10;
for i=1:N
    j = ceil(rand() * length(X));
    x_i = X(j);
    y_i = (10 - abs(x_i)) * rand;

    E(:, i) = [x_i y_i];
end

However, the points are not uniformly distributed, as clearly seen in the left and right corners. How can I improve that result? I've been trying to search for the different shapes too, with no luck.

Answer 1

You should first ask yourself what would make the points within a triangle distributed uniformly.

To make a long story short, given all three vertices of the triangle, you need to transform two uniformly distributed random values like so:

N = 1000;                    % # Number of points
V = [-10, 0; 0, 10; 10, 0];  % # Triangle vertices, pairs of (x, y)
t = sqrt(rand(N, 1));
s = rand(N, 1);
P = (1 - t) * V(1, :) + bsxfun(@times, ((1 - s) * V(2, :) + s * V(3, :)), t);

This will produce a set of points which are uniformly distributed inside the specified triangle:

scatter(P(:, 1), P(:, 2), '.')

Note that this solution does not involve repeated conditional manipulation of random numbers, so it cannot potentially fall into an endless loop.

For further reading, have a look at this article.

Answer 2

That concentration of points would be expected from the way you are building the points. Your points are equally distributed along the X axis. At the extremes of the triangle there is approximately the same amount of points present at the center of the triangle, but they are distributed along a much smaller region.

The first and best approach I can think of: brute force. Distribute the points equally around a bigger region, and then delete the ones that are outside the region you are interested in.

N = 1000;
points = zeros(N,2);
n = 0;
while (n < N)
    n = n + 1;
    x_i = 20*rand-10; % generate a number between -10 and 10
    y_i = 10*rand; % generate a number between 0 and 10
    if (y_i > 10 - abs(x_i)) % if the points are outside the triangle
       n = n - 1; % decrease the counter to try to generate one more point
    else % if the point is inside the triangle
       points(n,:) = [x_i y_i]; % add it to a list of points
    end
end

% plot the points generated
plot(points(:,1), points(:,2), '.');
title ('1000 points randomly distributed inside a triangle');

The result of the code I've posted:

one important disclaimer: Randomly distributed does not mean "uniformly" distributed! If you generate data randomly from an Uniform Distribution, that does not mean that it will be "evenly distributed" along the triangle. You will see, in fact, some clusters of points.