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I need to set a coordinate system fro a triangular grid so I did this:

X=========X=========X=====  \ -1, 0 / \  1, 0 / \   \     /   \     /   \    \   /     \   /     \     \ /  0, 0 \ /  2, 0 \ =====X=========X=========X     / \  0, 1 / \  2, 1 /    /   \     /   \     /   /     \   /     \   /  / -1, 1 \ /  1, 1 \ / X=========X=========X====== 

(The pattern continues in each direction)
Basically I converted the triangular grid to rectangular grid with the axe system:

+======> X | | V Y 

But another axe system can be used:

     X'    /   /  / x  \__     \__        \__ Y' 

Which is the same thing rotated 60 degrees counter clockwise, and is symmetric to the first because the triangles are regular = symmetric in 60 degrees.
This produces the following coordinates:

X=========X=========X=====  \ 1, -1 / \  2, 0 / \   \     /   \     /   \    \   /     \   /     \     \ /  1, 0 \ /  2, 1 \ =====X=========X=========X     / \  0, 0 / \  1, 1 /    /   \     /   \     /   /     \   /     \   /  / -1, 0 \ /  0, 1 \ / X=========X=========X====== 

That creates two coordinate systems. How can I convert a pair of coordinates $(x, y)$ in the first coordinate system to a pair of coordinates (x', y') in the second coordinate system?

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