How do i generate a rotation matrix iteratively.

I have two matrices containing coordinates:

example1 = [1 3; 
            4 5; 
            2 3; 
            6 17];
        
example2 = [1 4; 
            6 2; 
            8 9; 
            10 11];

and I'm implementing the RANSAC algorithm to remove outlier coordinates. To do that, I will need to include a factor of whether a not one coordinate in example1 is either a rotation, scaling or translation which result in the coordinate in example2. (1 3 -> 1 4, 4 5 -> 6 2 etc..) Hence, I will need to have a rotation matrix, scaling and translation matrix to be built iteratively within the RANSAC algorithm.

For now, I'll just keep it simple by just using a rotation matrix.

How do i generate a rotation matrix [cos θ -sin θ; sin θ cos θ] if i do not have anything else other than the two coordinate matrices above?

Edit: sorry for the confusion but the goal is to "estimate" the transformation matrix in every N iteration. The following shows:

Answer 1

[Walter Roberson]

example1 = [1 3; 
            4 5; 
            2 3; 
            6 17];
        
example2 = [1 4; 
            6 2; 
            8 9; 
            10 11];
        
syms theta
RM = [cos(theta) -sin(theta); sin(theta) cos(theta)];
residue = simplify( sum(sum((example2 - (RM * example1.').').^2)) );
best_theta = vpasolve(diff(residue,theta));

It is also possible to code it as a numeric minimization of residue over a bounded range, without using the symbolic toolbox.

[Stewart Tan]

@Walter Roberson thanks, but just in case i wasn't clear earlier, I've updated the post.

[Walter Roberson]

Your original goal was restricted to rotation matrix, and my code finds the rotation matrix that produces the least squared error of rotation between the given sets of coordinates.

[Walter Roberson]

I would suggest that Bruno's answer is probably better for your purposes.

Answer 2

[Bruno Luong]

You can use this implementation by Matt J using the Horn's method. It do for you the scaling and translation as well.