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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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