Uniformly-Distributed Points

Generate uniformly-sampled points in a 1D, 2D or 3D cartesian or spherical domain with optional pre-existing data via MAXIMIN design

Calling X = maximin(N, P, varargin) Generates N-uniformly distributed points in P dimensions using the following optional parameter-value pairs:

- 'iterations' Specifies the number of iterations, per cycle.

- 'cycles' Specifies the number of algorithm repetitions.

- 'criterion' Specifies 'cartesian' or 'spherical' sampling.

- 'data' Specifies existing data points in [M, P] form.

- 'initial' Specifies a set of [N, P] points to reiterate.

Examples of the use of this function are the following:

% Generate 40 cartesian equally-spaced points in 3D

x = maximin(40, 3);
plot3(x(:, 1), x(:, 2), x(:, 3), 'o')
view(3)

% Generate 20 equally-spaced points in 2D with pre-existing data

y = rand(10, 2);
x = maximin(20, 2, 'data', y);
figure
hold on
plot(x(:, 1), x(:, 2), 'x')
plot(y(:, 1), y(:, 2), 'o')

% Generate 12 spherical optimally-sampled points in 3D (icosahedron) and create edges

p = maximin(12, 3, 'criterion', 'spherical', 'cycles', 20);
figure
hold on
plot3(p(:, 1), p(:, 2), p(:, 3), 'o', 'Color', 'k')
for i = 1 : 12
    d = sum((p(i, :) - p) .^ 2, 2);
    d(d == 0) = Inf;
    [~, j] = mink(d, 5);
    for k = 1 : size(j, 1)
        x = [p(i, 1), p(j(k), 1)];
        y = [p(i, 2), p(j(k), 2)];
        z = [p(i, 3), p(j(k), 3)];
        plot3(x, y, z, 'Color', 'k')
    end
end
pbaspect([1, 1, 1])
view(3)

% Circular packing problem

p = maximin(10, 2, 'cycles', 20);
d = zeros(10, 1);
for i = 1 : 10
    j = sum((p(i, :) - p) .^ 2, 2);
    d(i) = min(j(j > 0));
end
d = 0.5 * sqrt(min(d));
a = 2 * pi * (0 : 1 / (1000 - 1) : 1)';
c = d * [cos(a) sin(a)];
figure
hold on
plot(p(:, 1), p(:, 2), 'x', 'Color', 'k')
for i = 1 : 10
    plot(c(:, 1) + p(i, 1), c(:, 2) + p(i, 2), 'Color', 'k')
end
xlim([min(p(:, 1) - d), max(p(:, 1) + d)])
ylim([min(p(:, 2) - d), max(p(:, 2) + d)])
pbaspect([1 1 1])