Distance between tiles in a hexagonal field?

I once set up a hexagonal coordinate system in a game so that the y -axis was at a 60-degree angle to the x -axis. This avoids the odd-even row distinction.

I assume that you want the Euclidean distance in the plane between the centers of two tiles that are identified as you showed in the figure. I think this can be derived from the figure. For any x and y, the vector from the center of tile (x, y) to the center of tile (x + dx, y) is (dx, 0).

The vector from the center of tile (x, y) and (x, y + dy) is (-dy / 2, dy*sqrt(3) / 2). A simple vector addition gives a vector of (dx - (dy / 2), dy * sqrt(3) / 2) between (x, y) and (x + dx, y + dy) for any x, y, dx, and dy. The total distance is then the norm of the vector: sqrt((dx - (dy / 2)) ^ 2 + 3 * dy * dy / 4).

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