August 16, 2024
Portals for pseudo non-euclidean spaces
https://www.bytehyve.com/view/blog/?i=84&t=portals_for_non-euclidean_spaces
If you start with the Euclidean space and warp it so that the X axis goes in a circle instead of an infinitely long straight line, you still have a space with Euclidean geometry. As a good analogy: you can take a piece of paper and cut it and glue it and obtain a cylinder. But you cannot obtain a sphere. So a cylinder is a Euclidean surface (can be made from flat paper) and a sphere is not. Cutting and gluing (as portals do) changes the topology, while, like stretching it changes the geometry. Same is you do that on all axes (as done in the game Manifold Garden for example, or Asteroids in 2D). You generally can have straight X and Y axes in spherical or hyperbolic geometry. What you cannot have is a square grid, or even a single square or rectangle with all edges straight and all angles 90°. Even a very small one. (In your videos there are lots of rectangles.) Non-Euclidean geometry is not really that complex math (CodeParade makes it look much harder than it actually is). If you already understand how to make 3D engines (linear algebra, homogeneous coordinates, projections, etc.) the math is not much harder. Of course there are less tutorials on it because less people are doing it. The effects you obtain with portals are very different from what you get from non-Euclidean geometry. I mean, technically, you can (2D analogy: you could do a polyhedron out of flat paper, for example, a 120-sided one, which is a good approximation of spherical geometry) but it is a very poor approximation and it is easier to do the real thing anyway.
I appreciate your take on it. Bending a single axis in 3D space still produces euclidean geometry with all axes running parallel. However, it can give off a non-euclidean feeling. A good example is Torus vs Sphere. A sphere has lines converging, unlike a torus. A torus resembles euclidean space, while the sphere does not. However, there is a major resemblance: every line/axes loop. With some slight trickery, you can make a torus look look/feel like a sphere. A great example is the game 'Eco (2018)', which has done this brilliantly. This means that yes, we are not recreating real non-euclidean geometry, but it may feel that way to the player. Thus impossible geometry, which may be the result of non-euclidean spaces, can possibly be recreated with some clever trickery. One way is with the use of portals. (Of course, not without its flaws) I believe you in the complexity of the math for a real non-euclidean engine (or lack of). However, it can still be a lot of work in comparison. Most players do not know much about non-euclidean spaces, meaning it does not have to be a perfect recreation to still be effective. (Still, a real non-euclidean game is extremely interesting)
Something that I want to add to this. My approach does not take non-euclidean space in the literal sense where all axes behave that way. It is more so specific lines, or specific spaces. Two parallel lines, one going through a portal and the other does not, do slightly behave in a non-euclidean way. Hence, creating the feeling of such impossible space. When done right, it can result in extremely interesting worlds. A great example is 'myhouse.wad (2023)', a Doom mod.
I have looked at Eco on Steam, the effect you are referring to it is not clear from the screenshots. But from your description, I understand that it is a torus (warped Euclidean plane) and it uses sphere inversion (or something similar) to make it look like a sphere. This trick has been used in some other games. This could be interpreted using non-Euclidean geometry -- it looks exactly like a horosphere in hyperbolic space. (A horosphere has intrinsic Euclidean geometry, just like a sphere has instrinsic spherical geometry.) It is not really related to the warping thing though. The horosphere itself is infinite. You could use the same trick to make infinite Euclidean plane to look like a sphere. (We have created a video "Non-Euclidean Third Dimension in Games" on this trick and some similar things that could be done BTW) I do not agree with "it may feel that way to the player" -- I see no overlap between how 3D hyperbolic geometry and a 3D torus feels. (Same for 2D.) It is interesting, sure, but just a very different thing :)
It is definitely very interesting. Great that you've made a video on the topic already which I'll definitely check out (for anyone interesed: https://www.youtube.com/watch?v=Rhjv_PazzZE). What I'm trying to explain in this blog post, is how geometry which is generally impossible in euclidic space can be recreated with some trickery. For example, the torus world which is most definitely not a torus. (btw, I'll revisit this topic later on, there are some interesting things possible with this technique) But yes. If you compare it to some weirder (trippy) non-euclidean spaces like hyperbolic spaces, it is different. I took a much simpler approach than you have done in the past. Something that I'll have a go at, at some point too.