How can I transform a coordinates data into another 3-D space?

suppose that I have coordinates of 4 points X1, X2, X3, and X4 which have form of Xi = [xi1; xi2; xi3; xi4] for i = 1~4 . Their relative position to each other never change. But the whole coordinates were rotated in some way, and now I measured the coordinates of X'1, X'2, and X'3 but not X'4. In this case how can I find the coordinates of X'4 in new 3D space? What function can I use?

+) If there is a linear transform T with 3x3 size matrix that satisfies T*[X1 X2 X3] = [X'1 X'2 X'3] then is it unique?

[Matt J]

If there is a linear transform T with 3x3 size matrix that satisfies T[X1 X2 X3] = [X'1 X'2 X'3] then is it unique?*

But you said your Xi are 4x1 column vectors. How can they multiply with a 3x3 matrix?

[Heejun Rho]

You're right. Xi's must be 3x1 column vectors

Answer 1

[Matt J]

You can use ABSOR (Download).

reg = absor([X1 X2 X3],[X'1 X'2 X'3]);

X'4= reg.R*X4 + reg.t;

[Heejun Rho]

This is exactly what I was looking for. Thank you very much for nice code and example!

Answer 2

[Mark Saad]

If you know angles through which they were rotated, you could try using rotation matrices.

https://www.mathworks.com/help/phased/ref/rotx.html

Answer 3

[Anton Semechko]

Code below demonstrates how to obtain rotation matrices between corresponding 3-tupples of non-collinear points in 3-space:

function three_point_rotation_demo

% Generate 3 random points and random rotation
% -------------------------------------------------------------------------
% 3 (noncollinear) data points 
Xo=randn(3,3);              % xyz-coordiantes along rows 

% Random rotation matrix
r=randn(3,1); 
r=r/norm(r);                % direction of rotation vector
t=(178*rand(1)+1)*(pi/180); % rotation amount; between 1 and 179 degrees 
r=t*r;                      % random rotation vector

K=zeros(3);
K(1,2)=-r(3); 
K(1,3)= r(2); 
K(2,3)=-r(1); 
K=K-K';                    % log of rotation matrix 
Ro=expm(K);                % rotation matrix corresponding to r

% Apply R to Xo to get X
X=(Ro*(Xo'))';

% Now, suppose that Ro (i.e., rotation that transforms Xo to X) is unknown. 
% -------------------------------------------------------------------------

% Compute local (orthogonal) reference frame for X 
d21=X(2,:)-X(1,:); 
d31=X(3,:)-X(1,:); 

e1=d21;                 e1=e1/norm(e1);
e2=d31-dot(e1,d31)*e1;  e2=e2/norm(e2);
e3=cross(e1,e2);        e3=e3/norm(e3);
F=[e1;e2;e3]'; % principal axes along columns

% Compute local (orthogonal) reference frame for Xo 
d21=Xo(2,:)-Xo(1,:); 
d31=Xo(3,:)-Xo(1,:); 

e1=d21;                 e1=e1/norm(e1);
e2=d31-dot(e1,d31)*e1;  e2=e2/norm(e2);
e3=cross(e1,e2);        e3=e3/norm(e3);
Fo=[e1;e2;e3]'; 

% Rotation we seek (R) satisfies the equation F=R*Fo. Since Fo and F are 
% orhogonal rotation matrices themselves, R=F*transpose(Fo)
R=F*(Fo');

% Now let's check that R is the same (up to numerical round off error) as
% Ro. If it is then R'*Ro should be a 3-by-3 matrix close to identity.
fprintf('Rotation residual:\n')
disp(R'*Ro)

% You can also verify that X=(R*(Xo'))', in which case, (R*(Xo'))'-X will be
% a 3-by-3 matrix with all entries very close to zero
fprintf('\nFrobenius norm of (R*(Xo''))''-X:\n')
disp(norm((R*(Xo'))'-X,'fro'))