How to compute a 2x2 Homography out of 3 corresponding points? In 1D projective geometry,
I want to compute the 2x2 Homography matrix $H$ (in homogeneous coordinates), given 3 pairs of corresponding points.
i.e. I want to find H such that:
$$\left(\begin{array}{cc}
h_{11} &  h_{12}\\
h_{21} & h_{22}\end{array}\right)\left(\begin{array}{ccc}
0 & a & a+b\\
1 & 1 & 1
\end{array}\right)   = 
\left(\begin{array}{ccc}
0 &a' &a'+b'\\
1 & 1 & 1
\end{array}\right).$$   
However, I've got 6 equations here and only 3 unknowns. 
(dof(H) = 4 elements less one for scaling = 3).
I thought about 3 scaling factors that would add up to 6 unknowns, s.t. we would have a unique solution. But how exactly do I insert the scaling factors into the matrices and how can I compute H then?
Do you have a clue?

Your answer is mathematically correct, however I figured out another way to solve this equation, which leads to a simpler result.
I applied the technique for 2D projective geometry, which is described here to the 1D case and it works out fine.
