[zkhbwvpj] Rectangular patch of a spherical manifold

In a 2D game, a character wanders around the surface of a large sphere.  The viewport is a square region centered on the player.  The region is small enough that there isn't much distortion.

If we maintain north as up, then things behave weirdly near the north and south poles: orientation of the viewpoint suddenly flips.

Better would be the player always faces toward the top of the screen, and the world rotates when the player makes a turn.  This might be especially useful on portrait display: one can see further straight forward than to the sides.  Center the player equidistant from the sides and bottom.  Icons or sprites of other objects on screen probably also rotate around their centers to remain right side up.  Circular icons would be best.

Inspired by the zillions of overhead-view 2D games in which the user sees just a section of a large world.

Providing an overview map of the whole world runs into the standard problems of map projection.

This could easily be generalized to any 2D manifold.

Perhaps the game helps in plotting the shortest path (geodesic) between two given points, e.g., player to target, when it is useful, because geodesics are not obvious on curved manifolds.  Or, it doesn't, and that's one of the mechanics making the game difficult.