identify a factor that links two equal matrices but positioned in different places on the same plane

I have two matrices ('RC_matlab' and 'RC_trasl') representing the same 2D coordinates of nodes on the same plane but at different plane positions.

I would like to identify the factor(s) that links the 'RC_trasl' matrix to the 'RC_matlab' matrix. That is, transform 'RC_trasl' as 'RC_matlab'.

I tried to determine the centroid (with a function found among the answers in the forum) of the two matrices in order to obtain dX and dY but this is not the correct way (apparently).

load RC_matlab.mat
load RC_trasl.mat

% polygoncentroid
[xc_matlab,yc_matlab] = polygoncentroid(RC_matlab(:,1),RC_matlab(:,2));
[xc_trasl,yc_trasl] = polygoncentroid(RC_trasl(:,1),RC_trasl(:,2));
dX = xc_matlab - xc_trasl;
dY = yc_matlab - yc_trasl;

% Plot
figure
plot(RC_matlab(:,1),RC_matlab(:,2),'b.','Markersize',30);
hold on
plot(xc_matlab,yc_matlab,'k*','Markersize',10);
hold off
axis equal

figure
plot(RC_trasl(:,1),RC_trasl(:,2),'r.','Markersize',30);
hold on
plot(xc_trasl,yc_trasl,'k*','Markersize',10);
hold off
axis equal

% New matrix
R_trasl_new = dX + RC_trasl(:,1);
C_trasl_new = dY + RC_trasl(:,2);
RC_trasl_new = [R_trasl_new , C_trasl_new];


% ===== function

function [xc,yc] = polygoncentroid(x,y)

    xn = circshift(x,-1);
    yn = circshift(y,-1);
    A = x.*yn-xn.*y;
    a = 3*sum(A);
    xc = sum((x+xn).*A)/a;
    yc = sum((y+yn).*A)/a;
    
end

Answer 1

[Voss]

load RC_matlab
load RC_trasl

% calculate the centroid of each set of points:
c_matlab = mean(RC_matlab,1);
c_trasl = mean(RC_trasl,1);

% calculate the offset of each point from its set's centroid:
d_matlab = RC_matlab-c_matlab;
d_trasl = RC_trasl-c_trasl;

% find the scale of the "matlab" set of points relative to the "trasl" set of points,
% by finding the mean ratio of the distances to their respective centroids:
scale_vector = sqrt(sum(d_matlab.^2,2))./sqrt(sum(d_trasl.^2,2));
scale = mean(scale_vector(scale_vector~=0 & isfinite(scale_vector)),1); % remove 0s, NaNs, and Infs, which can happen when a point is at the centroid

% then each  can be (approximately) constructed from the other by a linear transformation:
RC_matlab_recon = (RC_trasl-c_trasl)*scale+c_matlab;
RC_trasl_recon = (RC_matlab-c_matlab)/scale+c_trasl;

% Plot
figure
plot(RC_matlab(:,1),RC_matlab(:,2),'b.','Markersize',30);
hold on
plot(RC_matlab_recon(:,1),RC_matlab_recon(:,2),'g*','Markersize',10);
hold off
axis equal

figure
plot(RC_trasl(:,1),RC_trasl(:,2),'r.','Markersize',30);
hold on
plot(RC_trasl_recon(:,1),RC_trasl_recon(:,2),'y*','Markersize',10);
hold off
axis equal

% check the max difference between the original and the reconstruction:
max(abs(RC_matlab-RC_matlab_recon),[],'all')

ans = 1.0281e-04

max(abs(RC_trasl-RC_trasl_recon),[],'all')

ans = 5.0200e-05

[Alberto Acri]

Thanks for your answer @Voss.

I actually wanted to identify the factor or factors that connect the "RC_trasl" matrix to the "RC_matlab" matrix. That is, if I multiply (for example) the "RC_trasl" matrix with that parameter(s) I obtain the "RC_matlab" matrix as a result. If we look at the figures I posted: the red nodes must be moved towards the blue nodes.

I'm not clear on your solution :D

[Voss]

The scale variable is the relative scale; you can use that in conjuction with the centroids to transform from one to the other as shown in these lines:

RC_matlab_recon = (RC_trasl-c_trasl)*scale+c_matlab; % RC_trasl -> RC_matlab
RC_trasl_recon = (RC_matlab-c_matlab)/scale+c_trasl; % RC_matlab -> RC_trasl

[Alberto Acri]

Hi @Voss! I have a question for you, which was then the question of the post.

Is there a way to transform RC_transl into RC_matlab_recon? For example, a multiplicative parameter?

I would like to get something like this:

RC_matlab_recon = RC_transl*x;

[Voss]

"Is there a way to transform RC_transl into RC_matlab_recon?"

Yes, exactly as I said in my comment before (and demonstrated in my answer before that):

RC_matlab_recon = (RC_trasl-c_trasl)*scale+c_matlab;

You can't do it with only a scalar multiplicative parameter, but that expression includes a multiplicative parameter (scale). The expression is translating RC_trasl by its centroid (so that the points become centered at the origin), then applying the multiplicative factor (scale), then translating such that the centroid is the centroid of RC_matlab_recon (which is c_matlab).

You could do the same thing with a linear transformation matrix, and maybe that's what you have in mind. https://en.wikipedia.org/wiki/Transformation_matrix