[Freeciv] Re: world as a globe

["Ross W. Wetmore" <rwetmore@xxxxxxxxxxxx>]

At 08:03 PM 03/01/15 +0100, Reinier Post wrote:
>
>On Mon, Jan 13, 2003 at 03:42:18AM -0500, Ross W. Wetmore wrote:
>
>> One can map a hexagonal grid onto a sphere maintaining uniform 
>> distances and orientations.
>
>No way.  Can you explain further?
> 
>-- 
>Reinier

If you think of 0 and 4 as being the poles, and 2 the equator with
0-4 being some sort of latitude or distance from the pole, then a
fully expanded sphere looks like this where in fact the star 
shaped points represent the backside hemisphere. Note 4 is an
equivalent point (1/6th of a real tile, and 3s are half tiles
made by the flattening cut.

               4
              3 3
         4 3 2 2 2 3 4 
          3 2 1 1 2 3 
           2 1 0 1 2   
          3 2 1 1 2 3 
         4 3 2 2 2 3 4 
              3 3
               4

One would probably represent this as two hemispherical maps in the
following way as this gets all the wrapping/movement relationships
more or less right without too much brain strain. The <==> is meant
to indicate that the 2s along those mirror image edges are half-tiles
or equivalent positions.

             2 2 2   <==>    2 2 2    
            2 1 1 2  <==>   2 3 3 2   
           2 1 0 1 2 <==>  2 3 4 3 2   
            2 1 1 2  <==>   2 3 3 2   
             2 2 2   <==>    2 2 2     

I probably should have tipped each of these pi/6 so the vertical
axis, rather than the horizontal one was the long axis ending in
a point, and then marked the distances from the pole/points in 
rings from the top and bottom to make it even closer to what one
usually expects. But my ascii artist reflexes are weakening and 
the equator is not quite so obvious in this representation as 
the distortion in polar distances is subtly different with this 
orientation :-).

Such a "globe" can spin but there are only six representable views
along each of the three spin axes.

One might do better with a duodecahedron model (12-fold symmetry).

Cheers,
RossW
=====