Sunday, March 22, 2015

On the original hex grid guide I had claimed there were over 70 variants of hex grids. Most of these turn out to be merely a different choice of axes, either by renaming them or by negating the sign. Here are some variants of offset coordinates:

rename q and r offset axes

And here are some variants of axial coordinates:

rename q and r axial axes

Axial and Cube are really the same, except Cubes explicitly store the third coordinate and in Axial we calculate it in the accessor, when needed.

rename cube axes

Why would you want some of the other labelings? If I use an alternate labeling of offset coordinates, I can rotate the entire grid from pointy top to flat top and back:

rotate offset hex grid

This simplifies the math. It means I no longer need to treat pointy and flat top hexes separately, but instead I can focus on just a few basic grid types (cube, axial, even offset, odd offset) and then produce the pointy and flat top from those. I can further simplify by merging axial and cube together. I don't yet know if I want to do that. Can I merge even and odd offset? Yes, probably, but I think I won't right now.

parity-offset.png

Pointy top vs Flat top is a rotation. I had originally claimed it was a rotation of 30° but it is simpler to think of it as a rotation of 90°. It's even simpler to think of it as a permutation of x,y into y,x (renaming axes). There are also several coordinate systems that I don't cover on the page, and don't plan to anytime soon. However I might add a supplemental page that describes them.

I believe the new descriptions of coordinate systems will be simpler than the previous ones. However, they're also different, and I worry about changing the system that I've described in some incompatible way.

  1. Should I unify Axial and Cube?
  2. Even if I don't, should I rename Cube's coordinates from x,y,z to q,r,s? That way they match up more closely, and Cube is no longer confused with cartesian coordinates.
  3. Should I unify even Offset (even-q, even-r) and odd Offset (odd-q, odd-r)? I think the math is slightly uglier but it'd work just fine.
  4. Should I try to support all possible axis assignments, or just some "canonical" ones like I do now?
  5. Should I try to preserve the specific choices of axes on the current page, or choose the ones that make things simpler?

Update: [2015-03-28] I'm no longer convinced that swapping x,y is the easiest way to deal with pointy vs flat top. It's the simplest implementation, but it causes "q" and "r" to no longer correspond to "column" and "row", and I think it might be worth preserving that mnemonic. A 90° rotation is nice to implement too but I think the 30° rotation best preserves q/r.

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