Manhattan Distance between tiles in a hexagonal grid

Question

For a square grid the euclidean distance between tile A and B is:

distance = sqrt(sqr(x1-x2)) + sqr(y1-y2))

For an actor constrained to move along a square grid, the Manhattan Distance is a better measure of actual distance we must travel:

manhattanDistance = abs(x1-x2) + abs(y1-y2))

How do I get the manhattan distance between two tiles in a hexagonal grid as illustrated with the red and blue lines below?

Answer 1

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.

_{(source: althenia.net)}

The distance in this coordinate system is:

dx = x1 - x0
dy = y1 - y0

if sign(dx) == sign(dy)
    abs(dx + dy)
else
    max(abs(dx), abs(dy))

You can convert (x', y) from your coordinate system to (x, y) in this one using:

x = x' - floor(y/2)

So dx becomes:

dx = x1' - x0' - floor(y1/2) + floor(y0/2)

Careful with rounding when implementing this using integer division. In C for int y floor(y/2) is (y%2 ? y-1 : y)/2.