[Freeciv-Dev] Re: nonstandard maps

[Gaute B Strokkenes <gs234@xxxxxxxxx>]

On Sun, 12 Aug 2001, mkaufman@xxxxxxxxxxxxxx wrote:
>
> No. the mobius strip is the easy example. visualize a strip (or make
> one) and then think about the normal vector to the surface (color
> one side). That is your "up" orientation. If you follow the strip
> around you'll come to the point where you have a discontinuity in
> the Gauss map (the map of normal vectors) that is where you get
> screwed (it's actually worse than that because if you keep going,
> the side that you originally designated as "up" is now "down",
> that's why it's non-orientable).  You're (at least I am) sticking
> your terrain onto one side of the mobius strip (the side that's
> oriented "up"). When you come to the discontinuity, you basically
> hit the back side of your map (there ain't any terrain there) so you
> can't move in that direction period.

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
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.

I do know what I'm talking about; after all I intend to specialise in
algebraic topology.

>> > 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

  ^
 \/

close to the left of the bottom cross.  Now move up across border 4
into the upper cross and the left across border 1.  At this point your
twirl is

 ^
 \/

Or so I think.  It would help if you made explicit which "way" the
borders are to be joined.

     *->>-*
     |    |
     |  4 |
     |    |
     |    |
*----*    *----*
|              |
^ 1          2 ^
^              ^
|              |
*----*    *----*
     |    |
     |    |
     |  3 |
     |    |
     *->>-*


     *->>-*
     |    |
     |  3 |
     |    |
     |    |
*----*    *----*
|              |
^ 1          2 ^
^              ^
|              |
*----*    *----*
     |    |
     |    |
     |  4 |
     |    |
     *->>-*

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