Covering Earth with Hexagonal Map Tiles

carrier_20498 0
Viewed 56930

Many strategy games use hexagonal tiles. One of the main advantages is that the distance between the center of any tile and all its neighboring tiles is the same.

I was wondering if anyone has any thoughts on marrying a hexagonal tile system with the traditional geographic system (longitude/latitude). I think it would be interesting to cover a globe with hexagonal tiles and be able to map a geographic coordinate to a tile.

Has anyone seen anything remotely close to this before?

UPDATE

I'm looking for a way to subdivide the surface of a sphere so that each division has the same surface area. Ideally, the centers of adjacent sub-divisions would be equidistant.

hexagonal-tiles tesselation coordinates math
14 Answers

Well, lots of people have made the point that you can't tile the sphere with hexagonal tiles - maybe you are wondering why.

Euler stated (and there are lots of interesting and different proofs, and even a whole book) that given a tile of the sphere in x Polygons with y Edges total and z vertices total (for example, a cube has 6 polygons with 12 edges and 8 vertices) the formula

x - y + z = 2

always holds (mind the minus sign).

(BTW: it's a topological statement so a cube and a sphere - or, to be precise, only their border - is really the same here)

If you want to use only hexagons to tile a sphere, you end up with x hexagons, having 6*x edges. However, one edge is shared by each pair of hexagons. So, we only want to count 3*x of them, and 6*x vertices but, again, each of them is shared by 3 hexagons so you end up with 2*x edges.

Now, using the formula:

x - 3*x + 2*x = 2

you end up with the false statement 0 = 2 - so you really can't use only hexagons.

That's why the classical soccer ball looks like it does - of course modern ones are more fancy but the basic fact remains.

user_760510

It is impossible to cover a sphere with regular tiles (except for long and thin "orange slices". So the optimal way to pixelize a map, given certain constraints or requirements, is actually a pretty difficult research problem.

One sort of tiling used very often (in astrophysics) is the HEALPIX pixelisation: http://healpix.sourceforge.net/

This pixelization satisfies the equal-area requirement; it's impossible to make everything equidistant, however.

Another pixelization is "GLESP", which has some different properties (and isn't as polished a software package): http://www.glesp.nbi.dk/

user_122660
Kristo_468210

You can't cover a sphere with equal hexagons, but you could cover it with a geodesic, which is mostly hexagons, with 12 pentagons at the vertices of an icosohedron, and the hexagons slightly distorted to make it bulge into a sphere.

user_788400

Hexagonal tiles are too complicated for regular geometry as applied to geospatial uses. Check out HTM for a similar thing with triangles or google for "Hierarchical Triangular Mesh" for other sources.

user_792940

Read "Geodesic Discrete Global Grid Systems" by Kevin Sahr, Denis White, and A. Jon Kimerling

You can find it here...

user_2927390

Getting a sphere to divide into equal parts made with flat surfaces is a tough nut. Because of this, you end up with Geodesic shapes, which are not composed of shapes that can be in turn composed of triangles of equal size. Breaking down all of the hexagons and pentagons into triangles, you end up with triangles that have different interior angles, leading to a loss of symmetry.

The one consolation that I can give you is that all of the shapes will have a limited number of triangles that can be catagorised, which means for a small geodesic, that 5 or 6 triangles can be used repeatedly to describe all of the hexagons and pentagons required for the geodesic. While distances will not be equal from the "center" of each triangle/shape, you can at least divide the handling of each triangle into a discrete case, lending to a potential work-around in code.

user_869440

The old Traveller roleplaying game used to map planet surfaces as icosahedra (cut open for printing in a book). This produced a big distortion at the corner hexes (they have to become pentagons). You might find some such material when searching for GURPS Traveller.

Harleqin_316150

There is a paper which handles the case of equal area tiling (almost square tiles around the equator) and is relatively easy to pre-compute neighbouring tiles and on which tile a specific set of coordinates falls in. It doesn't fare well with the requirement for equal distance between the vertices though.

Copying here the abstract:

A new method is proposed to divide a spherical surface into equal-area cells. The method is based on dividing a sphere into several latitudinal bands of near-constant span with further division of each band into equal-area cells. It is simple in construction and provides more uniform latitude step between latitudinal bands than other methods of isolatitudinal equal-area tessellation of a spherical surface.

(I've used its ideas trying to find closest geolocation neighbours from a long list of locations).

user_68990
Related