Packing Circles inside a Circle - matlab

Question

Given unit circle, and a set of M smaller circles of radius r. Find the maximum radius of the smaller circles that allows them all to fit inside the unit circle without overlap.
I have the following circles packing in polygon example link
I want to change equations that say that all circles are inside the polygon
theta = 2*pi/N; % Angle covered by each side of the polygon
phi = theta*(0:N-1)'; % Direction of the normal to the side of the polygon
polyEq = ( [cos(phi) sin(phi)]*x <= cdist-r );
to equations that say that all circles are inside the circle, but i don't know how. Can somebody help me?
Kind regards.

Answer 1

In your case, there are no "sides of a polygon" so there is no analogue to theta and you'll need to change every place theta is referenced, and all variables that reference theta (like phi).
Something like the following should work. I've just copy-pasted the code from your link, gotten rid of theta and phi, redefined cdist and polyEq, and made it plot a unit circle in the answer instead of a polygon. Please ask if any of these choices are unclear.
M = 19; % number of circles to fit
toms r % radius of circles
x = tom('x', 2, M); % coordinates of circle centers
clear pi % 3.1415...
cdist = 1; % radius of unit circle
%%%%%% equations saying all circles are inside of unit circle
polyEq = (sqrt(x(1,:).^2 + x(2,:).^2) + r <= cdist);
% create a set of equations that say that no circles overlap
circEq = cell(M-1,1);
for i=1:M-1
circEq{i} = ( sqrt(sum((x(:,i+1:end)-repmat(x(:,i),1,M-i)).^2)) >= 2*r );
end
% starting guess
x0 = { r == 0.5*sqrt(1/M), x == 0.3*randn(size(x)) };
% solve the problem, maximizing r
options = struct;
% try multiple starting guesses and choose best result
options.solver = 'multimin';
options.xInit = 30; % number of different starting guesses
solution = ezsolve(-r,{polyEq,circEq},x0,options);
% plot result
x_vals = (0:100)./100;
plot(sqrt(1 - x_vals.^2),'-') % top of unit circle
plot(-1.*sqrt(1 - x_vals.^2),'-') % bottom of unit circle
axis image
hold on
alpha = linspace(0,2*pi,100);
cx = solution.r*cos(alpha);
cy = solution.r*sin(alpha);
for i=1:M
plot(solution.x(1,i)+cx,solution.x(2,i)+cy) % circle number i
end
hold off
title(['Maximum radius = ' num2str(solution.r)]);