[Freeciv-Dev] Re: (PR#7287) Extended Topologies

["Marcelo Burda" <NOSPAM_mburda@xxxxxxxxx>]

<URL: http://rt.freeciv.org/Ticket/Display.html?id=7287 >

Le lun 23/02/2004 à 17:37, rwetmore@xxxxxxxxxxxx a écrit :
> <URL: http://rt.freeciv.org/Ticket/Display.html?id=7287 >
> 
> 
> Just for amusement's sake, here is a recipe to make a map of real Earth
> in a torus. The projection to the 2-D torus-world surface is thus an
> easy second step.
> 
> 1)  Take a spherical Earth and stick a straw through the N-S polar axis.
> 
> 2)  Widen the diameter of the straw stretching all the latitudinal lines
>      correspondingly to maintain the hemispherical property of longitudinal
>      cuts. Once the straw diameter is greater than the original diameter
>      of the sphere you have a half-torus, namely the outer half.
> 
> 3)  Reflect the half torus through the cylinder walls of the straw and
>      rotate the inner half torus by 180 degrees.
> 
> Voila, one has a torus with a map of the real Earth.
> 
> 
> Note, a normal set is represented by a half torus, and there are two
> normal sets in a super-tile that contains the entire torus surface.
> 
> When one maps this to a 2-D torus-world, one usually does a standard
> Mercator projection, i.e. stretch the latitudinal lines as one moves to
> the poles, but shrink the longitudinal direction correspondingly to
> preserve constant area of any square patch. A cut along any longitudinal
> line allows one to peel and flatten the map out onto a 2-D plane.
> 
> Cheers,
> RossW
> =====
> 
> Marcelo Burda wrote:
> > Le dim 22/02/2004 à 15:48, Ross Wetmore a écrit :
> > 
> >>Jason Short wrote:
> >>
> >>><URL: http://rt.freeciv.org/Ticket/Display.html?id=7287 >
> >>>
> >>>rwetmore@xxxxxxxxxxxx wrote:
> [...]
> > Yes torus is easy, but not sphere like!
> > Can you make a map of real Earth in a torus? no.
> > Can you make a map of Earth in quincuntial? Yes.
> > Yes quicuntial is a litle harder to code. but i am coded it, pleas help
> > me to make it easy to undestand.
> [...]
> > Marcelo.
> 
Ok, i need refulmulate the question!
Can you make a good map?

You make a nice map-art but in you torus you create N europa! N america
etc. etc.
in a sphere there are One Europa, etc, etc.

Sorry, but torus can not canverted in sphere. there are not
equals(homéomorphes) topologies(math). Torus has a hole, sphere not!!!
(in math we separate the orineted clossed surfaces by the numbers of
holes: a sphere is a 0-Torus of R³, the torus is a 1-Torus of R³, there
are 2-Torus,etc. there are not equals!) 

If you make not single projection from sphere to torus. you can map over
any object. But this is not the some Topology(math).

To call 2 topologies equals you need at last find a single relation
point by point.(homéomorphisme)

This is wrong to thinck you can assimilate 4-Quincutial map to One torus
and say that is equals. in 4-Quincuntial map you See 4 time all units!! 

Torus(or twisted torus) are A clossed 2D space, sphere are a Clossed 2D
space! that is ok
But there are not equals!!!!!!!
there are not 1-1 point projection.

Quincuntial is A clossed 2D space and there are a 1-1 point
projection(and conformal) exept for 4 points. this is the best way to
project the sphere in a FLATE 2D if you like a clossed 2D space.

Why quincuntial as 4 singularitie? The sphere surface are not Flat, and
the Quincuntials surface are flat. This is compensated in the 4
singularities.

Best regards
-- 
 . /  .     '    ,    .      (*)   '        `     '      `    .    
  |    ,  |   `     ,     .      ,   '  Marcelo Julián Burda      .
 /  '     \     `     \@_     '      .        '      `        '    
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