Most interesting topology

[geeslaka]

cylinder: (currently used in freeciv) if a unit goes off the left side of the map it wraps around to the right side, and vice versa

 ________________________________
|                                |
|                                |
|                                |
|a                              b|
|                                |
|                                |
|                                |
|                                |
 ________________________________

a unit going left from point a would move to point b
a unit moving left from point ( x_min, y ) would move to point ( x_max, y )
toroid: (very common in RPG's) if a unit goes off the top of the map it wraps around to the bottom and vice versa, also any unit that goes off the right side wraps around to the left and vice versa

 ________________________________
|            c                   |
|                                |
|                                |
|a                              b|
|                                |
|                                |
|                                |
|            d                   |
 ________________________________

a unit going up from point c would move to point d
a unit going left from point a would move to point b
a unit moving left from point ( x_min, y ) would move to point ( x_max, y )
a unit moving up from point ( x, y_max ) would move to point ( x, y_min )
square: no wrap-around effects at all
mobius strip: indistinguishable from a cylinder unless there is some way for units to move through the surface, and the wrapping rules are different

 ________________________________
|            c                   |
|                                |
|     a                          |
|                                |
|e                              f|
|                     b          |
|                                |
|                            d   |
 ________________________________

a unit that digs a hole to the other side from point a would come out at point b
a unit that went up from point c would move to point d
a unit that went right from point e would move to point f
a unit that digs a hole to the other side from point ( x, y ) would come out at point ( ( x + ( map_x_size / 2 ) ) % map_x_size, ( y_max - y ) + y_min )
a unit that went up from point ( x, y_max ) would move to point ( ( x + ( map_x_size / 2 ) ) % map_x_size, y_min )
a unit that went left from point ( x_min, y ) would move to point ( x_max, y)
cylinder capped with frictionless hemispheres: a unit that goes off the map comes back on halfway around from where it started

 ________________
|   a       b    |
|                |
|e              f|
|    c       d   |
 ________________

If a unit went up from point a it would move to point b(and vice versa).
If a unit went down from point c it would move to point d(and vice versa).
if a unit wend left from point e it would move to point f
if a unit went up from point ( x, y_max ) it would move to point ( ( x + ( map_x_size / 2 ) ) % map_x_size, y )
If a unit went left from point ( x_min, y ) it would move to point ( x_max, y)
post any thoughts you have on these topologies and ideas for other topologies

Edit: typos and explanations
Tip: do not write explanations when you are tired
Edit: added formulas for figuring out where a unit ends up in the special cases

[Pave]

If a unit went up from point a it would move to point b(and vice versa). If a unit went down from point c it would move to point d(and vice versa). if a unit goes off the map at point (x,y) it comes back on at point ((x+(map_x_size/2))%map_x_size, y)

What?

[geeslaka]

Think of that rectangle as being wrapped around a cylinder. Now place a hemispere on each end of the cylinder. Now imagine that a legion is at point a. That legion moves up onto the hemispere. Since the hemisphere is frictionless the legion cannot stop on it. The legion slides all the way around until it hits the cylinder. The formula is the way to figure out where the legion ends up.

[geeslaka]

Another topology came to me in a dream last night. Take the formula that is applied to the top and bottom of the rectangle in the cylinder capped with frictionless hemispheres topology. Apply that formula to cases where units go off the left or right side as well and you get a much more complex topology.

 ________________________________
|         a               b      |
|h                               |
|                               c|
|                                |
|                                |
|g                               |
|                               d|
|           f               e    |
 ________________________________

If a unit goes up from point a it moves to point b.
If a unit goes right from point c it moves to point d.
If a unit goes down from point e it moves to point f.
If a unit goes left from point g it moves to point h.

[St Leo]

From a certain perspective, the Moebius Strip is what the current map already is.

[geeslaka]

Yeah. I wrote in my explanation for the moebius strip that it indistinguishable from a cylinder unless some special things are done.

[CapTVK]

I vote for the mobius strip. Okay, okay, gamewise an ordinary map plays the same but if you could teleport/warp to the opposite 'side' it would become something else. There's something appealing about playing on a map with that feature.

Would make for a great SF scenario.

[geeslaka]

I think(sometimes) there are four ways of implementing the ability to dig through the surface of the moebius strip:
(simplest first)[list=1][*] Add a terrain improvement that settlers/engineers can build
A settler can be told to build a tunnel on a space. When that tunnel is finished units can move through it to the other side.
Requires: 2 orders(one to tell the settler to build the tunnel, one to tell units to move through it) and 1 terrain improvent[*] Make all units able to move to the other side
Requires 1 order[*] Add a unit flag
In units.spec units could be flagged to indicate that they can move itself to the other side.
Requires 1 unit flag and 1 order[*] Combine 1 and 3[/list=1]
I favor 4.