The centroid of the boundary is the mean value of the coordinates:
x = rand(40, 1).^2; % More points on the left
y = rand(40, 1).^2; % More points on the bottom
k = boundary(x, y);
c = mean([x(k), y(k)], 1); % Center of points of boundary
plot(x, y, 'r.');
hold('on');
plot(x(k), y(k), 'b');
plot(c(1), c(2), '*g');
You see, that this is not the center of mass, but the centroid of points. To get the center of mass:
[CoMx, CoMy] = centroid(polyshape(x(k), y(k))); % Center of Mass
plot(CoMx, CoMy, '*k');
% or:
[cx, cy] = CenterOfMass(x(k), y(k))
function [cx, cy] = CenterOfMass(x, y)
% This fails, if lines of the polygon intersect.
x = x(:);
y = y(:);
x_ = circshift(x, 1);
y_ = circshift(y, 1);
A = x .* y_ - x_ .* y;
As = sum(A) * 3;
cx = sum((x_ + x) .* A) / As;
cy = sum((y_ + y) .* A) / As;
end
