Rendering Hyperbolic Spaces - Hyperbolica Devlog #3
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- Опубліковано 31 тра 2024
- This is the 3rd devlog for Hyperbolica where I talk about how I build and render hyperbolic worlds. If you haven't seen the first 2 videos in the series, make sure you watch them first or else this may be confusing.
Devlog #1: • Non-Euclidean Geometry...
Devlog #2: • Spherical Geometry Is ...
Hyperbolica on Steam: store.steampowered.com/app/12...
Trailer: • Hyperbolica: A Non-Euc...
Discord Server: / discord
Merch: crowdmade.com/collections/cod...
If you like the things I do and want to support the channel:
/ codeparade
ko-fi.com/codeparade
*CORRECTIONS:*
I just found out that Gyrovectors are indeed associative. I misunderstood the context of the claim in the paper, which was referring to mobius addition, not the full gyrovector addition which corrects the associativity. I apologize for the mistake!
Other references from video:
Miegakure: miegakure.com/
Ungar's original paper about Gyrovector Spaces:
www.researchgate.net/publicat...
Music:
Horrible - Mela
melagroup.bandcamp.com/album/...
gotta say, programming a true non-euclidean game without overuse of portals is itself a puzzle.
Once you have mastered thinking with portals... you must try... thinking without them?
Yup, I've been making a hyperbolic game as well. Something that trips me out as I'm programming is how messed the coordinate system is haha. I miss left, right, up and down, but they don't mean anything in this space!
yeah, this is the real big boy leagues
Are you telling me that you can walk forwards 19 meters.... without portals???
@@raconvid6521 at least its meters, I know those. it doesn't matter its a gyrovector made of 2 quaternions and 1 "thing" that's a complex number
In one of the levels you should place a large rock at coordinate "ULURU"
Y e s .
I t s h a l l a l l o w A u s t r a l i a n s o n l y t o
a p p r o a c h i t
Ha
...why?
@@ninjacat230 uluru is a big rock in the middle of Australia
@@jamiecasimir5040 yus
"If you haven't seen the first 2 videos in the series, make sure you watch them first or else this may be confusing."
Oh, don't worry, this is definitely still flying over most of our heads anyways...
I just watch it because it's trippy especially the spherical geometry one
only people that focused in school can understand this big brain stuff
*"And, though it was perhaps unsurprising that Cameron should be unenthusiastic about Juncker's candidacy, the vehemence of his opposition was extraordinary."*
I'm a mathematician, and when he started talking about Gyrovectors, my brain told me to go look that up. The Operation that's described in the paper he puts on screen is a fucking monster, as well. Although, it kind-of sounds like he's describing Quarternions, so "gyrovector" could just be a generalized term for the algebraic structure that quarternions operate in. Although, the non-associativity is definitely *not* something that Quarternions do, _and_ it presents a major problem for anyone who wants to do algebra with them, since inverse operations are difficult at best.
The most fascinating part of these videos is that despite my fear of these topics flying over my head, the explanations are so carefully presented that I still feel like I understand. This person is a great teacher.
Yeah, his animations about how walking a path in a different order leads you to different locations, or how moving can add unexpected rotation, really grounded the abstract rules of the gyrovectors in a neat way.
I think a great teacher should make the student actually understand the topics, not "feel they understand".
(While it is of course impossible to explain non-Euclidean rendering to a 5yo in five minutes, gyrovectors are not the simplest way to do it.)
CodeParade: Gyrovector translations aren't commutative
Me: Yeah, sure, I can deal with that.
Codeparade: ... or associative.
Me: _wait_
i had the exact same reaction
haha I don’t know what any of that means
@@tosterm "Not commutative" - the order of things matters. "A, then B" is not the same as "B, then A". Inconvenient, but manageable.
"Not associative" - the grouping of things matters. "A, then "B, then C"" is not the same as ""A, then B", _then_ C". Fuck that.
@@tosterm Commutative means you can change the order of the things you want to transform without changing the result, like 5 + 8 = 8 + 5 = 13, while associative means you can change the order of the operations themselves, like (7*2)*5 = 14*5 = 70 = 7*10 = 7*(2*5). Associativity is more fundamental than commutativity, so hearing that associativity stops working feels weird. (Sorry by any wrong words, English is a second Language to me)
@@lonestarr1490 But he's using them to model translations in hyperbolic space.
Translations are functions => their compositions is associative.
But gyrovector addition is not associative, so something doesn't add up.
“If you have not heard of ‘gyrovectors’…” when I in fact not heard of most of what he talks about, yet cannot stop watching.
This is literally the coolest game I have ever seen. I will be buying it as soon as it comes out!
It's this and teardown, my wallet will suffer…
Check out HyperRogue. It's a similar geometric concept, but the gameplay itself is kinda shit (in my opinion. I just don't like roguelikes.)
Me too (if I can)
And Meigakure as well...
same but kinda terrifying
I came for answers, and... boy did I get them.
So that’s why Miegakure’s been in development for so long? Yeah, I sure realize why now!
And I like how you reused your spherical geometry map, the one with the house and the fence, in demonstrating constructing hyperbolic geometry maps!
i'm so confused on all of this stuff.....
it really does explain a lot, in fact for me it actually kinda made me feel stupid about myself because the answer is so... SOO simple, just makes me think on how i didnt think of it
of course its been in develpment for over 10 years, go ahead and tell me a game engine that can run 4 dimensions
oh, well i'm actually relatively new to all of this stuff so it makes sense i'm confused. i like the videos though, they're cool(at least, the parts i can understand)
Noticed the Minecraft pig in your picture. Thought I should let you know I'm working on a hyperbolic minecraft game :)
@@kayturs Ayo??
This is top UA-cam result for Gyrovector.
It is at 4th now
This is one of the few things making me not hate my Linar algebra and Calculus classes anymore.
C'mon calculus is fun :)
If you want to hate it even less - I'd like to recommend you the channel called Insights into Mathematics or the playlist of algebraic calculus on another channel of him.
@@sillygoose210_6 for me math is fun once i understand it, but getting there is a real pain in the ass.
I love algebra and league ❤️.. calculus just a little 🤭
There is such a thing as "Tensor Calculus" :-P :-) and apparently tensors often contain a nested structure of lesser tensors rather than just being of scalars/reals/complex.
And here I am staring at my calculus problems for school and going “wtf why are there more letters than numbers?”
Jesus Christ, I’m looking forward to trying to play this game!
2:12 Getting flashbacks to making a Rubik's cube solver
We just need F and B to finish out the set
3d hyperbolic space let's go
but seriously, if you can map the "coordinate" of a 3d hyperbolic space onto any rubik's cube scramble, how close does that make them to being in the same group/set? (Sorry if I'm using the wrong terminology I'm not good with group theory yet)
Edit: Thinking about it, rubik's cube moves are also (I believe) not commutative or associative, the only hole would be inverse moves but if you re-wrote them into 3 normal moves it could totally get mapped to a hyperbolic space
HYPERBOLIC RUBIK'S CUBE
@@Supertimegamingify Oh god I wonder what THAT would look like
@@OrangeC7 What's tricky about this is there's no good way to add points on hyperbolic space together. In order to think of scrambles as points in hyperbolic space, you'd need points associated to 2 different scrambles to "sum" to a new point which corresponds to the composition of the 2 scrambles. That would mean the structure of the scrambles and their corresponding points would be the same. Unfortunately, there's no structure on hyperbolic space.
What you CAN do is map each scramble to a *transformation* of a hyperbolic space. These could be rotations, scalings, etc. Composing two transformations gives you a new transformation, just like composing two scrambles gives you a new scramble! However, To fully capture all the different ways to scramble a Rubiks cube, you'd probably need to consider hyperbolic spaces with more than 3 dimensions.
A similar line of thinking (mapping groups onto transformations of euclidian space) constitutes a whole field of math called "representation theory", which surprisingly enough is a vital part of modern physics!
Yay another devlog hyperbolica is epic.
big epic
Hyperbolica*
Yes. Keep it coming. Please. I beg you!
Big epic by hurting my brain xD
,
Why don't you just use Cinema 4D for the 4D game???
he knows too much
@@rehehehehehe4525 yes
How are you so wise with your scientific ways?
lol
maybe its because its very expensive? or maybe he just wanted to make a system from scratch lol could be anything. either way its really cool
Virtual Reality mode
This is already amazing, but I feel like these types of games are an even bigger boom in VR. And theoretically, I don't think you would need to change the rendering pipeline too much to add a VR mode. I'm not sure what type of user interactions you wanna add, but Unity has really good VR tools. And I really think it would boost the game's visibility, since there aren't many VR games yet, especially as unique looking as this.
And actually feeling the distance of things change would I think add a whole another dimension (literally) to the visuals.
I have a feeling that, considering motion sickness with just normal VR is a problem, our brains/inner ears would not be very happy in a VR setting viewing hyperbolic space
@@netsch20That is a valid concern, but I don't think it would be a big issue.
While a small percentage of people will never get used to VR and always feel sick a little, most of VR users get used to smooth locomotion and have 0 issues after some exposure. Me personally, I had 0 issues with smooth locomotion games right from the getgo.
And there already are some demos of 4D space and I haven't heard complains.
I think it's at least worth a try, I'm sure some (if not most) would be able to enjoy it. And like I said, I think even standing in spherical spaces would be mind blowing. Don't forget you actually see how far things are. I really think it would be an otherworldly experience. Something you literally cannot see otherwise.
@@netsch20 I have a feeling that that entirely depends on the person; some people get motion sickness on a hair trigger, some effectively don't get it at all
@@rarebeeph1783 Yes exactly. But most people get used to it. Only very few are unable to get used it.
I would totally love to see a VR mode, though possibly later in development. Once all the important stuff is out of the way, it would be a fun experiment to try and probably turn out better than expected! :D
When you realise that code parade is the first person to talk about gyrovectors on yt
Love this series. The perfect blend of Math and Code to produce literally a new perspective.
I love this kind of vlogs. You don't realize how much time consuming this is until you actually start doing it. Also, since there's not a lot of youtube videos on the matter, documenting this kind of stuff could be really helpful for someone trying to do something similar.
Really love how the game is coming along so far. For whatever reason, I kind of dig being disoriented in the hyperbolic space and having to relearn a lot seems kinda fun & a challenge. Looking forward to the final game, you're doing God's work son
You're the reason I've begun to lean forward building my own game, inspired avidly🥰
4:04 "I lead others to a treasure I cannot possess."
I didn’t realise it was so complicated to figure out how to walk around in hyperbolic space
me too. thought you'd just do some trickery with the output view
There is a way to store locations in non-Euclidean space. You just need to use metric tensors[ en.wikipedia.org/wiki/Metric_tensor ]. For a 3D space, the metric tensor would be 3x3. In your case, there's no curvature up or down. As such the metric tensor could probably simplify to a 2D-esque form[ en.wikipedia.org/wiki/Poincar%C3%A9_metric ]. Essentially, metric tensors tell you how much space is expanded or contracted in any given direction at any point. They can describe curved, non-Euclidean space extremely well.
In order to get the components of the metric tensor you still need a coordinate system, and a poor choice of coordinates will make the metric tensor components very complicated, so this doesn't avoid the coordinate system issue.
By the way, I think the game also has curvature up and down.
big brain man
YOU'RE A LIFESAVER, i've been trying to make my own hyperbolic game but i wanted to use a coordinate system. Couldn't find anything on google and this was the last place i had a chance of finding a way.
@@gigaprofisi You're going to want to research things like Riemannian manifolds, geodesics, and stuff like that. Also, the tile method used in the video would probably be described as using charts in GR. Essentially, you can carve up the manifold into chunks so that they have their own coordinate systems and if you walk off a chart, you walk onto a different chart. The benefit of that method is it allows you to define points in that region(like in a mesh) independently of how warped that region is in the full coordinate system. If you had a region which was highly distorted(like near the edge of the disc), then you wouldn't be able to precisely represent points there. You could apply a coordinate transformation such that the determinant of the metric tensor(volume element) was constant. This would let you represent points with equal precision everywhere, though they may be more or less precise along different axes. I'm currently taking a GR course BTW, so I'm not an expert. Feel free to ask more questions and good luck!
File format for 4d files
.dddd
(This is a joke but if I'm right it'd get even funnier)
We need to start a petition
What about .4do
I've been thinking. Objects in spherical space should look darker in the distance than nearby.
Because objects far away are stretched, they emit less light that reaches each of the pixels on screen. This means objects stretched 2x would look 4x darker because each pixel we would normally see would be spreading it's light across 4 pixels.
This would somewhat be counteracted by a different effect where pixels on object textures would be visible from bigger viewing angles. This would increase amount of light we would be able to see. So in spherical space every pixel on flat surface when viewed from maximum distance would be visible from 180 degree which would allow us to see every light ray it emits, but these light rays would also be stretched across entire sky, so the end result should be darker sky.
I think reverse should be true for hyperbolic space. More pixels would be firing light rays at us, but they would be doing so at smaller angle.
You're a pioneer to the next step in the evolution of gaming.
THE GAME "EVERYTHING" PEEK MY INTEREST INTO HYPERBOLIC SPACE. YOU GET TO VISUALIZE HYPERBOLIC SHAPES. EYE OPENING TO THIS DISCUSSION. THANK YOU.
This man's taking flat earth to whole new dimensions
"You can multiply these matrices together..."
Me, a mathematician: yes... 'Multiply'.
I believe the technical term is "smoosh".
@@OriginalPiMan Fusion is just a cheap tactic to make weak matrices stronger
This video is honestly fascinating. I love the depth to which you covered each concept while also not overexplaining; it's clear you've put a lot of work and passion into this project. I await more renditions and the game itself eagerly!
Don't know anything about maths, coding or gaming but I like very much the production value and your pronunciation (which many people lack, making their actually interesting videos crap to watch). Can't wait for Devlog #4!
there is a tile in the game called: "LUL"
Ok?
*L U L*
Left Up Left
I'd call that a DULL RUUL, obviously any combination exists
omegalul
I hope you put more architecture in the game! Like multi-floor buildings with confusing corridors, staircases, and walkways in and between them.
This geometry really lends itself well to that as opposed to just open environments.
Among the coolest and informative stuff I've seen this year. Thanks for explaining the Miegakure challenges, I've been checking out the progress several times a year.
I had numerics as an engineer... this is the E student watching the A student understanding and finding use of what I just accepted, memorized as algorithms and didn't find useful or interesting at all...
everyone: "with the exception of outer space, there is no such things like uncharted territories!"
CodeParade: "Hold my Euclidean geometry..."
Man, can't wait to blow my mind with Hyperbolica!!! Thanks for this logs, super interesting!
I think ill write my bachelors Degree about Hyperbolic Spaces. This is wicked!
this looks amazing man, probably the most unique game i’ve ever heard of, how has this not been mass invested in yet? non euclidean gaming, genius
This devlog inspired me to implement hyperbolic geometry in unity as a school project.
Have you also taken a look at what he made open-source from the game after it's release?
Absolutely Fascinating! Not only teaching me (a complete layman) something about game design and programming, but also the nature of the real world, and general problem solving approaches as well!
I'm so jealous of your ability to draw smooth lines without a ruler - I find it physically impossible! 😅
Wow you do such a good job of explaining these concepts. Thanks for making these dev logs, they’re very entertaining
"31 seconds ago"
yay
yay
I deeply congratulate you on that useless achievement
"hace un día"
2:06 the Euclidean grid is giving me learning Rubiks cube algorithms flashbacks
you are the ultimate answer to "professor, when are we gonna use these formulas in life?"
The fourth dimension and 4D games was a topic that I used to be interested in
Imagine this being an online game, and being able to play the game like a shooter in hyperbolic space
oh sh-
2:10 This is a well-studied problem. The magic phrase to Google is something like "reduced words in Coxeter groups" or "automata for Coxeter groups".
The way this whole thing suddenly connected to the Maths was beautiful.
You lose commutativity in hyperbolic space as you mentioned, which is why you have to use non-commutative quaternions (eg 4:24) which then directly connects to the non-commutative group behind quaternions.
It's really nice how flawless geometry, algebra and group theory come together in this topic.
2:11 The tiles answer each other because RU/UR sounds like Are you/You are
Thank you for sharing your work! This is fascinating stuff. I loved your 4D video, and I look forward to seeing more videos! :)
The position representation you have chosen is (perhaps somewhat unintuitively) actually quite efficient. A bit simplified, you need an amount of information that is logarithmic to the number of squares you want to be able to represent in order to encode the position of a single square, and since the area of a disc grows exponentially to the radius in hyperbolic space, you need an amount of information that is linear to the distance from the origin to encode the position of a square, which is exactly what you have used! Good work! (One way to "compress" the representation of all squares would be to store them all in a tree, or perhaps a graph for convenience, which would only require a constant amount of information for each square, but I guess you have already figured this out.)
that rendering is pleasing to the eye
Can’t wait to see more videos on this game. It’s such an interesting concept
I love the idea of this game and i admire all the effort you're putting in this, i'll surely buy it.
P.s: This series definitely needs more views
This video applies to more than just euclidian spaces. I often have to work with the problem of lack of tools in the area of music interfaces, having to decide to start from scratch or use existing tools to skip time. I love this short video, it shows your way of thinking really clearly!
A massive multiplayer game in hyperbolic space where you can claim a plot to build a house on: everyone can have one, and yet still be so very close together!
After spending 2 hours in Unity to make a TWO DIMENSIONSAL character move, I'm all the more impressed by your dedication. Take your time, we'll be patient!
> I couldn't even find a single video about it
I wasn't expecting such a jumpscare in this devlog. Shouldn't you warn us about it?
Man, I got headac just by looking on this parabola,
that is something to be proud of that you can actually develop this kind of stuff!
Man this could really be a big step in videogames. Imagine the FC4 drug missions with this kind of stuff.
I get a feeling that you have all the tools and ideas at your disposal to solve all the problems of current day Virtual Reality.
I’m imagining a hyperbolic shooter and I think that would be dope
I like that you try to use already avaiable tools as much as possible. It's efficient.
Wow. you explained the position matrix so good in a few seconds that i understood more than in my programming class in which we had to work with it a bit and just got some code
i dont code or do this advanced level of maths but this is fascinating
Woo! I'm so glad we get to watch the development, it definitely makes it seem like it's not going to go the way of Miegakure and stall out.
I don't care if the end product doesn't even feel like a "game" but rather a tech demo, I'll be shelling out to support this kind of advancement. Keep up the good work, loving these devlogs :)
2:42 "Just use the game itself." ---- Yes, please. I've had so much fun in Cube 2: Sauerbraten (and frustration), and wish all games were like this.
This game is gonna be a 1 of a kind when it comes out, and is inspiring a lot of people including myself.
But 1 question, is there going to be a deep story or will the game focus more on the Hyperbolic world aspect?
It will do both :)
One of the applications I’m anticipating is VR. If you have a finite play space to move around in, then unless you’re using expensive physical tech like slip shoes or omnidirectional conveyor belts, you’ll have problems navigating a space larger than your play space. Teleportation works but it’s clunky.
But if you folded your huge world into a compact form using clever hyperbolic geometry, in theory you could navigate huge open worlds within a relatively compact space. I’m sure it would have some complications and take a lot of getting used to, but in principle the game engine could map long linear distances in-game into spirals in the play space.
Sure it would have its limits, but to me there’s such a huge frontier I’m just excited to see what people do.
The idea of using spirals may not be hyperbolic geometry. However, that sounds like an interesting idea!
I would go with something else. What about... circular? From a different persons perspective, you will be walking in a circle. Inside VR, you are going straight, and not coming back to where you started.
Lev Peslyak The reason I said spiral is because a circle would map to an infinite straight line, but some kind of spiraling curve would eventually terminate against the boundary of the play space
@@syzygy6 I realised that what I described could be a horocycle in-game, mapped to a circle IRL.
If we include changing the radius of the circle you are walking along, you could explore an infinite corridor: limited width, but infinite length.
Either way, the width is limited while length can be made infinite.
Or do you know a way that allows both axis to be unlimited in a limited play-space?
I think you are confusing some things here.
You can indeed use portals to simulate huge worlds in a relatively compact space. This has been suggested in an older video by CodeParade, and is currently used by some VR games, e.g., Tea for God.
However, these are *portals*, which are sometimes incorrectly called "non-Euclidean geometry", but do not actually have anything to do with actual non-Euclidean geometry (or hyperbolic geometry in particular).
Actually, this trick no longer works if you are trying to simulate a non-Euclidean geometry, at least if you are copying the movements exactly. This is because, in a non-Euclidean simulation, you can move your head in a circle, and your real head will stay in place, but your virtual head will spin (because of the differences between the two geometries). So you can just move your head to make the orientation of your real and virtual head completely independent, so avoiding virtual walls will no longer guarantee that you won't hit a real wall.
ZenoTheRogue Portals are not “incorrectly” called non-euclidean. Portal geometry is every bit as non-euclidean as hyperbolic or spherical geometries, it’s just constructed in a more ad-hoc fashion. As long as it does not obey Euclid’s five postulates, it counts.
I’m mostly speculating on possibilities, and I suspect that a useful solution will rely on a combination of techniques. Fitting an open game world into a compact play space is currently an unsolved problem. Developing hyperbolic engines will probably bring us closer to a solution.
im beyond excited about this! keep up the amazing work, i cant wait to see what sorts of things can be done with hyperbolic/spherical space in games and this project is my favorite to follow rn bc of it
I remember seeing your noneuclidian worlds engine video a couple years ago then not long after forgetting your channel existed entirely, only finding you again after your previous upload to this one, your work is continually exciting and impressive, and your videos are quite interesting. They fill me with ideas of things I want to try making, and might even take a crack at once I'm further along with existing projects.
This is bloody clever stuff, mind my english, i think i learned more about 4-dimensional physics watching this video, than i could from less practical examples of non-euclidean geometry etc.
@SArpnt You are right but 3D non-euclidean should be isomorphic to some surface in 4D.
unlike yandev this guy has a reason its taking 10 years, i was born when this was beginning development
The 4d game has taken 10 years. This guy is doing a completely separate project, but he brought them up because both projects face similar issues. This project will hopefully be much quicker for the reasons mentioned in this video.
Brilliant devlog! It's very interesting to see the game and hear your development explanation.
Thanks for the update. I'm really excited by this project!
Code parade: Working on a single game for 10 years
Me and my lazy ass: I need this kind of work ethic in my life
I'm pretty sure the 4D game is being developed by someone else, but same
@@robhillen8007 oh.. I didn't know that, but the point still stands
@@robhillen8007 Correct. marctenbosch.com/news/
Can’t wait for full game and walkthrough, I wonder if there will be a boss or if this is a puzzle game? 🤔
This hurts my brain in a way that made me just wishlist this game. I've played some non-euclidean space games like antichamer which is probably the most notable example, but this is awesome in a whole different way. Less confusing and more mindbending in a fantastical way. It's super weird and out there, but after looking at it for like 30 seconds, my brain "gets" it, and it no longer feels too outlandish. That combination of amazing mathematical concepts and how our brains interpret them is way too cool, and thus, this is a game I'll have to experience.
Thanks for the introduction to gyrovectors.
Can we visit such a debug level in the game? This could enable us to learn more about how hyperbolic and spheric geometries work.
Ok, let's throw aside the technical stuff for a min,
Is there a story to this game? And if so, what would it be?
At every video you make, I just get more and more anxious to see the final job! Congratulations!
This video taught me more about hyperbolic geometry in 6 minutes than virtual classes have taught me in 5 hours.
i don't even know what half of these words mean, 10/10 would get confused again
this feels like I just barely learned the entirety of quaternions
I wish you luck. That's absolutely fascinating!
Holy crap! 5:18 is the first time I understood why it is called hyperbolic geometry.
gyrovectors do be like that sometimes
I would LOOOOOOOVVVVVVE if someone made a hyperbolic zelda game 😱😱😱
Yeah, new devlogs.
I haven't watched it yet but I expect it to be cool and hard to comprehend as usual (in a good way).
This devlog is very useful for lots of things - applying existing tools is a great skill since the code that you use is tested by others before you
4:44 Is a tensor product a subset of gyrovectors?
"It's hard to draw 4D scenes on 2D paper"
Then maybe 4D scenes could be drawn on 3D paper.
Your monitol is still 2D tho
@@Ascyt no mine is ULURU
such concepts are often really simple to grasp, and hard to master let alone implement. There is simply too much details and subtleties. Good job for your work, it looks beautiful. You must be proud of yourself!
i love your channel so much and am so very glad that i found it through the non-euclidean world engine video! thank you for all the amazing insights!
Q:
4:30 Rotations can't be performed in any order, can they? I may have misunderstood the difference in that property then if it doesn't work for normal rotation matrices.
Rotation, no. Translation, yes.
@@weckar understood
ayyy
Fascination! Intrigue! Me being way too tired to really process this but it simmering nicely all the same!
This is my favourite on going series on UA-cam. I absolutely cannot wait to play this game