# Change system of coordinates, having initial and final coords, find the way

I have points ( how many we needs) with coordinates $(x_1,y_1)$ , $(x_2,y_2)$ , $(x_3,y_3)$ ... in the cartesian System 1 ( actually, a millimetered sheet ) and the corresponding coordinates $(a_1,b_1)$ , $(a_2,b_2)$ , $(a_3,b_3)$ in another system, System 2, that we can assume cartesian.

E.g. from manual measurement I know points :
A is (334,491) in System 1, and ( 46.604856, 34.833369 ) in System 2
B is ( 1273,209 ) in System 1, and ( 46.609281, 34.835422 ) in System 2
C is ( 1721,1032 ) in System 1, and ( 46.608634, 34.840995 ) in System 2

Surely are involved a rotation, a linear deformation and a translation. I must find them, to calculate the coordinates of an arbitrary but near point ( near to my known points ) in a system, given the coordinates in the other.

I read Change from one cartesian co-ordinate system to another by translation and rotation. getting few hints, my level is far below this problem

so with three points and their images, you have $6$ linear equations and can solve for these 6 variables, $p,q,r,s,t,u$. One way would be Gaussian elimination on a $6\times 6$ (sparse) system of linear equations. Another would be writing this as a matrix equation, similar to what I did in this answer:
So if you have the formula for the inverse of a $3\times3$ matrix, a bit of matrix multiplication will determine the parameters of the transformation. Bear in mind that if the three lines are collinear, you have insufficient information and will encounter a division by zero.