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How to find the original symmetry group of a displacement map from a coordinate transformation?
A matrix $M$ and a point $A$ in the plane, both with homogeneous coordinates $u,v$, form a line $L_M: u=Mv.$ A displacement map $A\to T(A)$ from the line $L_M$ to the line $L_{T(M)}$ with displacement vector $C$ will be given with homogeneous coordinates $w,x$ as
$w=M^{ -1}ux,\qquad x=v.$ Now I want to know how $C$ can be found from $T$ and $M$.
This question is connected to my previous question, which is not answered as I wanted.
A:
This is the Lie algebra of ${\mathrm{SL}}_2(\mathbb R)$: the elements $M$ and $C$ form a basis, and we define the Lie bracket as
$[M,C]=C$
Now we can define the matrix of the matrix of the linear transformation $T$ with respect to the basis $M$ and $C$ as
$ M' = M - C. $
Using this and that $[M,C]=C$ we can calculate the matrix of $T$. In particular, the $2 \times 2$ minors of $M'$ give us $T$.
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