[xoukpkha] global curvature of square fundamental polygons

a square (or rectangle) fundamental polygon is good for 2D games.

previously on the various spaces (manifolds) that a fundamental polygon can represent, depending on how its edges are defined to behave.

do weird things happen at the corners of the square fundamental polygon for an objects larger than a point (e.g., a sprite) moving around the space?

sphere: Y
torus: N
real projective plane: Y
Klein bottle: N
cylinder: N
Moebius strip: N
hemisphere: Y
square: N

things are not weird if the neighborhood around corners behaves like flat (Euclidean) space.  investigate by tracing a small circle around each corner.  in flat space, we expect a 360-degree circle.  in the answers above, Y = (weird things happen at corners; there is global or total curvature; small circles are less than 360 degrees); N = (corners behave like flat space; circles are 360 degrees).

if you have a fundamental polygon with more edges than a square, e.g., hexagon, you can probably get small circles larger than 360 degrees, some sort of hyperbolic space.  you might be able to get some corners with positive curvature that cancel out other corners with negative curvature for zero global curvature.  but it would still be awkward for games, because local curvature is what we care about for rendering sprites.

even though weird things happen at corners of sphere, projective plane, and hemisphere when drawn as a square fundamental polygon, they are all manifolds (locally Euclidean), so there are other ways of depicting them so that small circles remain approximately small circles throughout, smearing the curvature or weirdness throughout the surface.

or, just put an obstacle over the weird points.