Converting Global Coordinates to Character Local Coordinates and Back

I am trying to implement obstacle avoidance behavior from the paper steering behaviours for autonomous agents. What I am stuck at is how do i convert global coordinates (2d) to local coordinates for my character?

Basically Say I am at 1,0 and the enemy is 10,0. I would like to move the origin to 1,0 so I get 9,0 as the enemy coordinates.

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What I ended up doing,

to translate to local,

[1 0 -tx]   [x]
[0 1 -ty] x [y]
[0 0  1]    [1]

then back to global using,

[1 0 tx]   [x]
[0 1 ty] x [y]
[0 0 1]    [1]

tx,ty is the local char coords and x,y is the enemy char coords.

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Just subtract the points.

Relative = Position - moved axis point.

(10,0) - (1,0) = (9,0)

Displacement

Edit:

Using an Affine transformation to convert the entire x,y plane:

Just for numeric issues, lets choose another perspective point: (3,7)

[x] = [ 1 0 -3 ] = [x`] = x -3
[y]   [ 0 1 -7 ] = [y`] = y -7
[1]   [ 0 0 1  ] = [1 ] = we don't care

Transformation matrix