Given 3 points (x and y coordinates) in coordinate system A, and 3 corresponding points in coordinate system B, how can I derive an AffineTransform that will convert from A to B.
My question is similar to Create transform to map from one rectangle to another?, except that question only deals with 2 points - i.e., it assumes there is no rotation.
Suppose your transform is of the form
x' = px + qy + r
y' = sx + ty + u
and write your six points as (A1x, A1y), (A2x, A2y), (A3x, A3y), (B1x, B1y), (B2x, B2y), (B3x, B3y). Expressing this in matrix form gives
/ \ / \ / \
| B1x B2x B3x | | p q r | | A1x A2x A3x |
| | = | | | |
| B1y B2y B3y | | s t u | | A1y A2y A3y |
\ / \ / | |
| 1 1 1 |
\ /
Now find the inverse of the 3x3 matrix on the right. You'll find plenty of algorithms online telling you how to do this. There's one at http://www.econ.umn.edu/undergrad/math/An%20Algorithm%20for%20Finding%20the%20Inverse.pdf, for example.
Post-multiply both sides of the equation above by the inverse of the 3x3 matrix, to get the values of p, q, r, s, t, u, v.