Is there a spherical equivalent to the euclidean/asteroids torus

[Bohandas]

Ok, so in videogames like the classic arcade shooter Asteroids the shape of the map is technically considered a torus. It's not curved, but the way things wrap around is equivalent to if you rolled the map into a cylinder and then bent the cylinder into a donut shape.

What I want to know is, is there a spherical equivalent to this? And if so what?

I can think of a couple of things that woulf be kind of like this, but I don't know if any of them would count. One would be doing the same sort of wraparound on a circular screen, another would be to cap the cylinder in the earlier example so that going off one rim of the cykinder takes you to the opposite side of the same rim instead of the same side of the opposite rim, and a third would be to just take a regular map projection and say that things that go off the edge of one part land on the other corresponding edge (although this has the disadvantage of no longer being square). Do any of these work? Are any of them close?

[crayzz]

You wanna look at topological quotient spaces

https://en.m.wikipedia.org/wiki/Quot...ace_(topology)

For a square specifically, you'd consider the edges of the square to all be the same point on the sphere

I think your cylinder example also works if I'm reading it right

[halfeye]

The thing about the torus is that it just works, very easily, because if you roll up a sheet of squared paper it makes a cylinder, then put the ends together and you get a torus (you can't actually do that with the paper, but the maths sort of works). It's not clear which way the torus is oriented to the screen, could be horizontal or vertical (but not any sort of diagonal or semi-diagonal).

If you try to map a sheet of squared paper to a sphere, it doesn't work half so well, all the squares get squished and pulled out of shape, they're a bit out of shape on the torus, but the right angle corners are still all right angles.

The one with the same side on the cylinder you have to think about the middle, if bottom goes to top, something that goes off the top corner has to do something along the top as well as at the bottom, and that's not obvious, and something going off at the middle has to come on again at the middle, and it will tend to hit itself coming back.

With the torus something going off the middle comes back in the middle of the other side, and that's okay, something going off the corner in the torus comes on in the diagonally opposite corner, and that's okay too. With anything else you have to work out something to deal with the middles and corners, and when it doesn't just work, it's not going to be easy.

If you want a sphere for some purpose, amd need to cover it in regular shapes, that's more or less possible, but it's not going to easily map to a rectangular screen. Lines of longitude and latitude are one way, another is to take the soccer ball shape (https://en.wikipedia.org/wiki/Truncated_icosahedron) and break the pentagons and hexagons into triangles, then you can make hexagons from the triangles, you'd be left with small pentagons in the middle of the big pentagons, but there aren't many of them and you could fudge something if you really hated pentagons.

[DavidSh]

The first construction, using a disc rather than a square, gives you the projective plane. If you move an extended object through the rim, it comes back through the other side in the opposite orientation, so a "p" turns into a "q". If this were an RPG map, to the characters, the world would appear to be inverted left-to-right.

I don't recognize the second construction, but I think it gives the same issue with orientation.

And yes, if you have a map projection that covers the entire globe, you can use that. Note that Mercator never reaches the poles at any finite distance. For this purpose I like using stereographic projection of each hemisphere into its own disc. Then when you depart one disc you enter the other disc at the appropriate point.

[Chronos]

The first two constructions are in fact topologically equivalent, and both are, as DavidSh says, non-orientable: A left-handed person could turn into a right-handed person by traveling around the world (this is the same idea as a Möbius strip).

The third construction, as I understand it, would just be another torus, with a (distorted, because it's no longer a sphere) world map painted onto it.

What exactly are you looking for with a "spherical equivalent"? You can get spheres involved if you're willing to go to higher numbers of dimensions. Mathematically, a torus is a two-dimensional space that's a circle crossed with another circle, but you could also have a three-dimensional space that's a sphere crossed with a circle, or a four-dimensional space that's a sphere crossed with a sphere.

[Bohandas]

I don;t think that's right. Wouldn't the projective [lane be if you connected it to the opposite end of the opposite rim rather than thenopposite end of the same rim

[DavidSh]

I don;t think that's right. Wouldn't the projective [lane be if you connected it to the opposite end of the opposite rim rather than thenopposite end of the same rim

I've looked this up, and find:
1) There are two topologically distinct ways to connect the top rim to the bottom rim of a cylinder. One way gives you a torus, the other way gives you a Klein bottle.
2) Connecting the points on a rim to their opposite points on the same rim gives you a structure called a crosscap. Doing the same process on both rims independently also gives you a Klein bottle.
3) Connecting the points on one rim to their opposite points while leaving the other rim open gives you a projective plane with a hole in it. You can repair the hole by either contracting it to a single point, or squashing it flat and taping it up.

[Bohandas]

I'm not gluing a m0bius strip or a whitney umbrella to it. I;m attaching an ordinary disc to each end and then shrinking the size of the disc to zero.

Once I put the two discs on it's definitely topologically equivalent to a sphere, and it seems it should remain equivalent because I'm only squishing and stretching after that.

The question is whether it would be a servicible sphere in any other context.

EDIT:
At that point I just have two cones with distorted grids on them, don;t I?

[Lord Torath]

I'm not gluing a m0bius strip or a whitney umbrella to it. I;m attaching an ordinary disc to each end and then shrinking the size of the disc to zero.

Once I put the two discs on it's definitely topologically equivalent to a sphere, and it seems it should remain equivalent because I'm only squishing and stretching after that.

The question is whether it would be a servicible sphere in any other context.

EDIT:
At that point I just have two cones with distorted grids on them, don;t I?

I was going to say you have a line segment, or a cylinder with zero radius. If you don't insist on maintaining constant radius (implied in a cylinder), then yeah, you've got a sphere or a cone, depending on how you stretch the profile.

[monomer]

You could break down the sphere into a large number of toruses (tori?) with decreasing radii until the radius at the edge of the screen is 0. To keep everything on a rectangular surface, objects would grow as they moved left or right until they are the height of the screen at either edge at which point they're just too big to keep going.

[Bohandas]

Ok, apologies. I just realized that I should have specified that I'm considering the case of something walking over this surface, rathet than embedded in it.

EDIT:To clarify the distinction here, consider if I have a paper mobius strip and a Hotwheels car. I can roll the Hotwheels car along the mobius strip for as long as I want and it's never going to turn into a British car with the steering wheel on the right

[Rydiro]

Topologically, you get a sphere, if you identify all the edges of the square into a single point.

[Chronos]

On a paper Möbius strip, the car will never turn into a British car, but it will turn into a car with the steering wheel on the right. It's just that, at the same time, it'll also turn into a car with its wheels on top. In other words, it'll be turned upside-down, and on the "other side" of the strip ("other side" doesn't have a global meaning for a Möbius strip, but it does have a local meaning).

[halfeye]

Are we supposed to be mapping a sphere onto a flat monitor / TV screen? I thought that was what we are doing, but this mobius strip stuff and the edges mapping to one point isn't something that can be made to work on a screen in any easy way, an image changing size as it approaches the edges is going to be really compute intensive, it might not be possible to do that in real time.