Consider a game with the following mechanic: one is given a 3D shape, an origin point, a destination point, and an off-geodesic point all on the surface of the shape. When traveling the shortest path from the origin to the destination, is the off-geodesic point to the left or right of the path?
Best is with a physical model. Globe and 3 cities for a sphere. Other shapes could follow a similar principle: a physical model with labeled locations (maybe regions labeled by unique shape or color pattern) allows the player to locate (by eyeballing) points that are displayed on screen on a zoomed 2D map projection. A graphical solution, if wanted, is given also on the display in map projection.
Purely electronic is trickier. Easiest is to fix the shape to be a box, and put rectangular display panels on the 6 faces. For an arbitrary fixed shape with a center (probably convex), an electron gun and phosphor surface seems good for 3 points and the solution line, in the style of old oscilloscopes. Other methods of projection from the center, e.g., laser, would also work. Previously, spherical display. The ultimate is a shapeshifting physical display, essentially Odo or the T-1000 Terminator.
Virtual reality (VR) and augmented reality (AR) could offer nice interfaces.
More complicated games: players complete to trying to draw a path shorter than those drawn by other players. Perhaps in the style of a racing game. Also previously, billiards.
Inspiration was complicated but doable algorithms for computing geodesics on a triaxial ellipsoid.
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