[MUD-Dev] Re: Hex-grid mapping

[James Wilson]

On Tue, 01 Dec 1998, Jon Leonard wrote:
>On Tue, Dec 01, 1998 at 07:51:24PM -0500, James Wilson wrote:

>> this isn't a hex grid. It's a warped square grid. A hex grid has the useful
>> property that any cell has six neighbors, which is often a better fit for
>> tactical simulations than the square grid's four-with-four-diagonals neigbors.
>> For instance, diagonal motion makes a lot more sense in hexes than in squares
>> (assuming one move equals the traversal of one border). 
>> 
>> You can't tweak your equation to give them six neigbors, so don't try. (The
>> number of neighbors of each cell is a topological invariant, i.e. you can warp
>> space in any smooth way and each square will still have
>> four-with-four-diagonals neighbors.)
>
>Would you have been happier if I'd drawn my hexes like this?
>
>        / \   / \   / \   / \
>       /   \ /   \ /   \ /   \
>      | 0,2 | 1,2 | 2,2 | 3,2 |
>      |     |     |     |     |
>     / \   / \   / \   / \   /
>    /   \ /   \ /   \ /   \ /
>   | 0,1 | 1,1 | 2,1 | 3,1 |
>   |     |     |     |     |
>  / \   / \   / \   / \   /
> /   \ /   \ /   \ /   \ /
>| 0,0 | 1,0 | 2,0 | 3,0 |
>|     |     |     |     |
> \   / \   / \   / \   /
>  \ /   \ /   \ /   \ /
>
>Those are really hexagons...  And the six neighbors have the coordinate
>deltas I specified.  The fact that they don't look pretty in ASCII graphics
>says little about the math.
>
>That they're best stored in memory as a two dimensional array doesn't change
>how the geometry works.  Two of the "corner" diagonals are as adjacent as the
>"sides", and the other two diagonals are farther away than with a square
>geometry.
>
>The underlying mathematics are that if you try to do coordinate geometry with
>axes 60 degrees apart, the distance function changes to d=sqrt(x*x-x*y+y*y),
>and the lattice points wind up being the centers of packed hexagons.

nevertheless there are eight neighbors of each node, not six. perhaps he doesn't
care about that.

James