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 Post subject: Re: Sonic 3 special stages are impossiblePosted: Wed Oct 18, 2017 8:54 am

Joined: Sun Jan 22, 2012 12:03 pm
Posts: 6806
Obduction had a really interesting topology.

The game takes place inside a sphere, and going through one side of the sphere wraps to a point opposite it. Of course, if you did this for all points in your body at once, you would be inside out after passing to the other side... so the game actually just treats you as a single point. It's a very interesting sleight of hand.

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 Post subject: Re: Sonic 3 special stages are impossiblePosted: Wed Oct 18, 2017 9:23 am

Joined: Wed May 19, 2010 6:12 pm
Posts: 2749
Is Super Mario Galaxy the only game that has Topography that makes sense?

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 Post subject: Re: Sonic 3 special stages are impossiblePosted: Wed Oct 18, 2017 10:57 am

Joined: Sun Jan 22, 2012 12:03 pm
Posts: 6806
tepples wrote:
An octahedron is isotopic to a sphere. By ripping four edges adjacent to one vertex, it becomes isotopic to a square interior. You go off (say) edge b, and you come out at edge b', from which it was ripped.

Specifically just mapping a square to a sphere isn't a big problem. There are all sorts of ways to do that; the cylindrical projection is probably the most popular method. The problem is dealing with a grid that you're not really able to distort, and might be impractical to change the orientation too.

Rendering this octahedron scheme as a 2D JRPG map would involve having the world appear in 4 different rotations depending on where/when you cross those edges. Around the midpoints of those edges things would get really confusing, like you'd be able to see a place diagonally across from its 180-degree twin on the same screen. I don't think it'd be very intuitive for the player, even if it technically could map to a "spherical" world. The distortion and loss of orientation here is a problem.

From a rendering standpoint, it'd be usable for Sonic 3's bonus stages, maybe, since it's a rotating perspective anyway? However, I think for level design it'd probably be a real pain to deal with the way it wraps, not to mention confusing for the player trying to figure out why in some places they can turn left 2 times and end up where they started.

I've seen a cylinder world in games a few times. Civilization IV, for example, blocks the north and south pole with a wall of ice, and otherwise lets it wrap horizontally. Keeps the nice flat grid, as well as the idea of a circumnavigable earth, but in addition to losing the ability to travel across the poles, the north and south extremes are also far too wide, the spherical shape it not being represented. Always therer is this conflict between keeping an oriented grid and not distorting its shape vs. the sphere.

psycopathicteen wrote:
Is Super Mario Galaxy the only game that has Topography that makes sense?

The earliest game I can remember playing that had an actual no-compromises spherical world map was X-Com: UFO Defense (1994). There's probably earlier examples, though.

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 Post subject: Re: Sonic 3 special stages are impossiblePosted: Wed Oct 18, 2017 11:51 am

Joined: Fri Nov 12, 2004 2:49 pm
Posts: 7517
Location: Chexbres, VD, Switzerland
rainwarrior wrote:
tepples wrote:
An octahedron is isotopic to a sphere. By ripping four edges adjacent to one vertex, it becomes isotopic to a square interior. You go off (say) edge b, and you come out at edge b', from which it was ripped.

Specifically just mapping a square to a sphere isn't a big problem. There are all sorts of ways to do that; the cylindrical projection is probably the most popular method. The problem is dealing with a grid that you're not really able to distort, and might be impractical to change the orientation too.

Rendering this octahedron scheme as a 2D JRPG map would involve having the world appear in 4 different rotations depending on where/when you cross those edges. Around the midpoints of those edges things would get really confusing, like you'd be able to see a place diagonally across from its 180-degree twin on the same screen. I don't think it'd be very intuitive for the player, even if it technically could map to a "spherical" world. The distortion and loss of orientation here is a problem.

But isn't mapping a 2D square to a torus basically the same issue? Warp around would be nice, but the squares would have to be extremely distorted for this to work at all.

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 Post subject: Re: Sonic 3 special stages are impossiblePosted: Wed Oct 18, 2017 12:18 pm

Joined: Sun Jan 22, 2012 12:03 pm
Posts: 6806
But isn't mapping a 2D square to a torus basically the same issue? Warp around would be nice, but the squares would have to be extremely distorted for this to work at all.

Putting a 2D toroidal map onto a 3D torus will necessarily distort its grid, simply because of the difference between the inner and outer radii. (I'm about 75% sure you can do it without distortion in 4D though.)

However, it is able to map the square without cuts or seams. You don't have part of one side connected to a backwards oriented other side; the whole thing is smooth and translatable across the entire surface. You can't say that about ANY mapping of a square to a sphere. There's always a cut/seam somewhere. You can stretch out a big flat area to make the seam tiny, or you can try to hide it in some impassible place, but there has to be one somewhere. At some point down will suddenly become up, or right becomes left, etc.

Edit: okay now I'm 100% sure you can do it without distortion in 4D because there's an article about it here: Clifford torus "The Clifford torus is an example of a square torus, because it is isometric to a square with opposite sides identified."

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