Last winter, while stepping onto a bus, I had an idea for an Asteroids-like game that took place in the Poincaré Disk, a model of the hyperbolic plane best known for its use in M. C. Escher tessellation art. Well, I haven’t followed through on that, because it would be hard to do. Instead, I grabbed a much lower-hanging fruit which, to a layman observer, has a similar visual effect.
Thus was Asteroidal Projection born, an Asteroids-clone that takes place in normal Euclidean space, except with the plane compressed into a disk-shape. Distances become increasingly distorted as you get farther from the center of the screen, with distant objects remaining visual, though warped, in the edges of the disk. This visual distortion gives the illusion that the game is occurring on a curved, dome-like surface, but this is actually not the case. Mathematically, the game world is topologically equivalent to the plane, and all the physics and collision detection occurs identically to how it would on a flat play area. You can think of the game’s visuals as an alternative way of viewing flat space which keeps much more of the play area in view at once but does not preserve shape, distance, or angle.
For the topology/complex analysis nerds out there, the visuals in Asteroidal Projection are the image of the Euclidean plane through this homeomorphism:
And yes, strictly speaking, this is not a projection, just the standard homeomorphism from the plane to the disk. What can I say? I didn’t think “Homeomorphic Asteroids” had the same ring to it.
Anyway, you can play it online here:
This “beta” is very informal, just drop me a line if you have any feedback! (firstname.lastname@example.org)
PS I still haven’t decided whether I’d rather use really minimalist, purely geometric visuals for this game. What do you think?