How it changes the the resulting figure?

Hello.

I wrote a code that makes a 2d embedding from the columns of P to the first 2 non trivial eigenvectors and works fine.

Now I need to recreate this embedding but after subtracting the mean of all columns of P. I tried the following

but I know that it recreates the same figure as my first one and verified that. So how subtracting the mean should affect the embedding?

theta=0:2*pi/1023:2*pi; 
t=-1.5:3/511:1.5;
P=shepp_logan_proj_2(theta,t);
C=(1/size(P,2))*sum(P,2);
X=zeros(size(P,1),size(P,2));
for i=1:size(P,2)
    X(:,i)=P(:,i)-C;
end
epsilon=0.5;
W=zeros(size(P,2),size(P,2));
for i=1:size(P,2)
    for j=1:size(P,2)
        W(i,j)=exp(-norm(X(:,i)-X(:,j)).^2/(2*epsilon));
    end
end
a=(sum(W,2))';
D=zeros(size(W,1),size(W,2));
for k=1:size(W,2)
    D(k,k)=a(k);
end
A=inv(D)*W;
S=D^(-0.5)*W*D^(-0.5);
[U,G]=eig(S);
V=D^(-0.5)*U;
c=sort(diag(G),'descend');
cc=1/size(P,2)*sum(V,2);
figure(1);
hold on
clf;
scatter(P(:,1),P(:,2));
xlabel('\itx');
ylabel('\ity');
axis equal;
axis tight;

figure(2);
clf;
scatter(V(:,2),V(:,3),5,theta)
xlabel('\phi_1')
ylabel('\phi_2')
set(get(gca,'XLabel'),'FontSize',12)
set(get(gca,'YLabel'),'FontSize',12)
axis equal;
axis tight;

Thanks in advance!