[Freeciv-Dev] Re: nonstandard maps

[Mike Kaufman <mkaufman@xxxxxxxxxxxxxx>]

On Mon, Aug 13, 2001 at 01:28:50AM +0200, Gaute B Strokkenes wrote:
> On Sun, 12 Aug 2001, mkaufman@xxxxxxxxxxxxxx wrote:

> This is where you are wrong.  To put it in Freeciv terms, whenever you
> add or subtract map.xsize from *x in normalize_map_pos(), you have to
> flip the y coordinate.  Off course you can no longer talk about things

It is extremely unclear what you mean be "flip".

> such as "north" or "south" in a meaningful manner, but that doesn't
> prevent you from having a well-defined map.
> 
> It helps if you think of the Möbius strip like this:
> 
>  
>   *----*
>   |    |
>   ^    v
>   ^    v
>   |    |
>   *----*
> 
> rather than considering an embedding in R^3.

That's difficult since the strip _is_ a surface in R^3...

The bounding curve of the surface is _not_ the issue (but even if you used the 
"right-hand-rule", you'd still run into probelms). If that were the case, how 
would you define the orientation of the surface of a sphere? From "Differential 
Geometry of Curves and Surfaces" by Manfredo P. Do Carmo, 1976, p.135:

  It is a striking fact that not all surfaces admit a differentiable field of 
unit normal vectors _defined on the whole surface._ For instance, on the Mobius 
strip of Fig. 3-1 one cannot define such a field. This can be seen intuitively 
by going around once along the middle circle of the figure: After one turn, the 
vector field N would come back as -N, a contradiction to the continuity of N. 
Intuitively, one cannot, on the Mobius strip, make a consistent choice of a 
definite "side"; moving around the surface, we can go continuously to the 
"other side" without leaving the surface. 
  We shall say that a regular surface is _orientable_ if it admits a 
differentiable field of unit normal vectors defined on the whole surface; the 
choice of such a field N is called an _orientation_ of S.

(This book kicks total ass by the way)
> 
> I do know what I'm talking about; after all I intend to specialise in
> algebraic topology.

yes, but geometry is different from algebra :P
> 
> >> > I can dream up all kinds of orientable surfaces that would be fun
> >> > or at least interesting to try out, but it would be nuts to code
> >> > or maintain the code.
> >
> >> I can't.  We are restricted by the need to be able to present the
> >
> > Here's an easy one: take two rectangles with flaps and join them
> > together at the ends of the flaps (number to number).
> >
> >          3
> >        XXXXX
> >        XXXXX
> >    XXXXXXXXXXXXX
> >  1 XXXXXXXXXXXXX 2
> >    XXXXXXXXXXXXX
> >        XXXXX
> >        XXXXX
> >          4
> >
> >          4
> >        XXXXX
> >        XXXXX
> >    XXXXXXXXXXXXX
> >  1 XXXXXXXXXXXXX 2
> >    XXXXXXXXXXXXX
> >        XXXXX
> >        XXXXX
> >          3
> >
> > I think that that one would be fun. (this sort of thing would work
> > for any n-gon n>2)
> 
> It's 2AM in Norway right now, but are you sure that's orientable?
> Imagine starting with a twirl

It's 7:30pm in Kansas now, and I'm not quite sure what the twirls are supposed 
to be :), but I'm pretty sure that's orientable, see the definition from 
DoCarmo above (of course, if we want to get technical, these surfaces can't be 
bounded or they wouldn't be differentiable, but that's immaterial for this 
discussion).

If this doesn't convince you, perhaps we ought to take this off-list. But as 
far as I'm concerned, this isn't the interesting part of my post. I want to 
hear your opinion of my spherical mapping proposal...

-mike
> --
> Big Gaute                               http://www.srcf.ucam.org/~gs234/
> Hello.  I know the divorce rate among unmarried Catholic
>  Alaskan females!!