Deriving 3D Rigid Body Physics and implementing it in C/C++ (with intuitions)

[@blackedoutk]

As some of you have pointed out, I made a few minor mistakes while talking about the math. I want to summarize them in this comment. First, at 26:06 I inserted the rewritten vector triple product into the integral without putting parenthesis around the expression. So in every following expression, it looks like rho(r) is only multiplied with the right addend, when it actually must be multiplied with both of them. However, if you imagine that there are brackets around the sum expression or think about rho(r)dV as being the differential mass element of r, which is what I did and how this mistake even occurred, then everything is still correct. 26:06 TL;DR: brackets missing visually, maths still handled correctly (as if they were there) Secondly, at 26:15 I moved some parenthesis and said that this was possible because matrix multiplication is commutative. I meant to say associative of course. Matrix multiplication is not commutative in general. 26:15 TL;DR: is used a wrong word Thanks to @bartoszstyperek6306 and @jaborl mentioning these and sorry for any possible confusion. If you find further errors, I would appreciate them being pointed out as a reply to this comment, so they are easier to find. UPDATE 2024-06-30 As @notu483 pointed out, the explanation prior to the conversion step done at 7:41 is a bit vague. If you want to do this more formally and correctly, I suggest to use a function w(x, t) to have a better distinction between the resulting time dependent world position w and the local position x that is being integrated. Then you can use Leibniz integral rule (specifically the one for higher dimensions aka Reynolds transport theorem) to do this conversion. Thanks for mentioning. 7:41 TL;DR: only important if you're interested in the specific mathematic details

[@blackedoutk]

@@matthiasmax2849 Hey, I feel what you're saying. Unfortunately, I'm not really sure what to recommend. I didn't read one specific thing that taught me all of this, rather I picked up some things here and there. Are you studying computer science? That would definitely be a good starting point. I feel like what helped me was working through exercises and writing down derivations step by step, even if it felt kind of pointless. And I would still recommend to attend the available simulation lectures, regardless of whether you have all the necessary prior knowledge. If that is possible. I picked up most of the math I present in these videos because of the animation and simulation lectures. I felt damn hopeless in the beginning, but because I was so interested in this topic I learned much by googling the little things or watching a few niche KZbin videos. Also, don't be afraid to ask your professors stuff. By the way, I also have a discord server now if you're interested, the link is in the comments of my second livestream. We could discuss this further over there. Man, do I sound that german 🙈I hope my english is still okay to listen to haha

[@PerriPaprikash]

probably best to fix the error in the video and then to reupload the video.

[@blackedoutk]

@@PerriPaprikash in some way true but unfortunately that kills the momentum of the video. and I would lie if I said that I don't care how many people I reach with these videos. after all I make them so people watch them. so that's a bit of a dilemma. additionally the comments would be lost

[@Archimedes.5000]

@@blackedoutkyou could have an unlisted fixed video, though hard to tell how would that affect the algorithm

[@terrastudiosdev]

FINALLLY!!! a new physics engine video 😀

[@blackedoutk]

yayyyyy

[@jackfrederiksen7979]

I want to understand this video, yet my calculus knowledge is limited to a highschool-level course. Please remind me in 2-3 years to come back and rewatch this video so I can actually understand all the math

[@blackedoutk]

I created an entry in my calendar in 2 years:D see you on the other side 😤

[@lanchanoinguyen2914]

you don't need calculus yet,you must be good at linear algebra and geometry.Programming is not easy as mathematical imagination,one logic goes wrong and the rest of the system will be ruined.

[@w花b]

​@@lanchanoinguyen2914 Imo it's easier, you can be a bit less strict unlike math that's unforgiving.

[@friedrichmyers]

@@lanchanoinguyen2914 Math is harder, bro. Programming deals more with Raw Logic, though.

[@warguy6474]

​math is 100% harder yeah@@friedrichmyers

[@Kraypus]

It's always interesting seeing these types of videos, which are always rare to come across without obnoxious over-the-top narrating, I'm glad I subscribed :)

[@blackedoutk]

That's nice to hear :)

[@jellewestra]

Mate, your videos are amazing; a great balance between math and implementation! Thank you and keep posting :)

[@blackedoutk]

Thank you, that's great to hear :) I feared there was too much math in this video

[@charliearmour1628]

I can almost understand the math, like an itch you cannot scratch. Love it. Great video, thank you so very much.

[@blackedoutk]

:D thank you. maybe I can help out? which part is not exactly clear?

[@omniopen]

I’m a mechanical engineering student and I didn’t really understand the whole inverting/ shifting axis of rotation thing until I watched this, excellent work!

[@blackedoutk]

that's so great, thank you!

[@andrewsearns5634]

I’ve been working on my own physics simulation engine as an opportunity to brush up on my understanding of physics as well as practice coding some more interesting projects. This video is phenomenal! The intuition presented here is so much easier to make sense of than everything else I’ve seen for handling 3D angular momentum. Part of the simulation that I’m interested in is handling a dynamic model that does not stay as a rigid body, but if I’ve internalized this intuition correctly, I should be able to use conservation of angular momentum combined with time dependent inertial tensor calculations to accurately model rotations in my simulation.

[@blackedoutk]

That's great, thank you so much. I am not exactly sure if this is applicable in your case, I mean the total angular momentum of the dynamic model must stay constant, but that is not true for any part of it, since they might exchange energy.

[@williamchurch8401]

This is over my head but I’m trying to learn it. Your videos are gold for helping me make that leap. Thank you! I am very much looking forward to the next one. Please don’t stop! Your efforts are greatly appreciated!

[@blackedoutk]

Thanks, glad to hear that :) I would recommend working through some exercises to get more comfortable with these concepts. Or even just writing down the derivations and trying to understand each step.

[@williamchurch8401]

@@blackedoutk Will do! I"m brushing up on my calculus as well.

[@tempname8263]

Try to check my comment up above, maybe it'll help?.. Maybe not

[@lucassamuel6069]

Amazing video! I was looking for something like this for a long time

[@blackedoutk]

Same :D Thanks!

[@gamedevgerry]

OMG! THIS MAN UNDERSTANDS ME! He legit just described my whole life in the first 3 min😂. Instant like and instant sub

[@blackedoutk]

hahaha very nice. the pain is real. thank you

[@aidankemp-harper2559]

I feel like ive been waiting for this video to exist my whole life. I think I can die happy now.

[@blackedoutk]

haha:D

[@FengXingFengXing]

Can use Monte Carlo method for calculate mass and inertia moment, mass center, etc for complex geometry one time (or when start game) and save values in code. Also Runge Kutta method for time if method is fast enough.

[@stevemcwin]

I really love the quality of the video. The explanation is also really good. These types of tutorials give me motivation to challenge myself with programming projects I've never tried before. I hope you continue making videos!

[@blackedoutk]

Thank you, that‘s nice! My goal is to make these things both easier to understand and implement. I hope I will continue too 😁

[@bobfake3831]

this vid is also a nice introduction to the actual physics of rigid bodies, well done

[@blackedoutk]

thank you :)

[@ballmathieu0]

legendary fyp pull

[@kafial7776]

aint getting no "we can go gyatt for gyatt" reels

[@blackedoutk]

damn I feel old, but thanks I guess haha

[@blackedoutk]

@@kafial7776 😂😭

[@MrHeinzelnisse]

Nice! I'm trying to wrap my head around the XPBD paper and I newly discovered your videos, amazing work so far, can't wait for the upcoming videos! Next thing for me will be to get constraints working with rigid bodies, I'll most likely mess it up and then wait for you to explain it :D

[@blackedoutk]

Thank you! Happy to hear that :D Unfortunately it'll still take quite some time until I get there. If I could, I would pump out these videos weekly, but that's just not doable for me

[@MrHeinzelnisse]

@@blackedoutk That's totally understandable, with these kind of topics I value quality over quantity, watching a bad tutorial that makes you more confused than before is not a good experience. That's definitely not the case with your tutorials, so keep up the good work!

[@Nonsense116]

What an incredible presentation. Taught in such a way I alllllmost feel like I could've gotten there myself. Such is the mark of all excellent explanations as well as the excellent teachers behind them. Excellent job and thank you for this gem!

[@pauluslinschoten8023]

Its just amazig how you build a sense out of the math. Thank you kindly.

[@cooper6506]

This video is AMAZING. This is a great intuition for inertial tensors and I finally understand them.

[@nagisa1578]

Thanks man, finally a video that not just scratches the topic of rigid bodies. After reading through a paper, watching this video makes me understand what I have read better. Thanks!

[@blackedoutk]

Glad it helped :) which paper did you read?

[@mani_mincraft]

legend! finally an in-depth dive into rigid body simulations (not that I can understand the math but still)!

[@blackedoutk]

ty! 😅 if you have questions regarding some parts feel free to ask. maybe I can help out

[@Lucas-pj9ns]

thanks for the well explained rigidbody physics

[@drjankenstein]

i have never understood how integrals work until now, but this made my brain finally grasp the continuous function pat

[@blackedoutk]

great to hear:D I really dislike it when there is a mathematical concept that I can't get an intuition of

[@samdavis649]

@@blackedoutk I really agree with this. Im studying physics at university, and find the maths im taught so much easier to grasp than the maths taught at alevel, strictly due to it being taught by physicists who naturally give iniuitive examples.

[@sashangovender5327]

I remember reading those witkin and baraf papers 20 years ago. They're the best at explaining how to implement rigid body dynamics.

[@Bekir_ts]

yessss my calculus course is worthy now !!!!

[@blackedoutk]

haha finally, right

[@NikHem343]

Me going into this video: "I don't know a lot about math but I'm a reasonably smart guy. On a surface level this should make sense." Me after 3:40: "I might have gone too far in a few places."

[@Theo-iz5cj]

Hey! This video just now popped up in my recommended videos. It was really good and I watched it all, understanding most I'd say. It has really helped me. Thanks a lot! I will be watching the next one(s) too.

[@blackedoutk]

Hey, thanks for your kind feedback. Feel free to ask about parts you didn‘t exactly understand, maybe I can help out

[@Speed001]

Very nice, i can see how statics, dynamics, and some coding/matrix knowledge is used. And the first half of a transcendental calculus textbook should be sufficient for anybody looking to learn.

[@blackedoutk]

thanks. that's useful to know, I always struggle a bit telling people what prerequisites they need

[@nartulga-jl7ug]

This is golden! thanks for making this video

[@blackedoutk]

Thank you :)

[@grippgoat]

I think I learned something from this. But I'd have to watch it several more times to tell you what. 😂 I appreciate you explaining the inertial tensor, because everything else I've read/watched has assume it's just a priori knowledge. With your explanation, I now have a bit of understanding of why unconstrained rigid bodies wobble and flip.

[@101arrowz]

My high school physics classes only taught rotational motion using moment of inertia as a scalar quantity - in hindsight, I think we only considered situations where an object was rotating about one of its principal axes. Learning about the matrix representation was a little mind bending :) Thank you for creating this excellent resource! I always thought physics engines in games used a bunch of hacky estimations to maximize performance, but your implementation is both mathematically grounded and seems relatively easy to optimize. Now I feel the urge to write one too...

[@blackedoutk]

Thank you for your kind comment :) I remember seeing the formulas for rotations about one principal axis in the formula book, but we never talked about it in high school unfortunately. Though we learned a bunch of other interesting stuff. I guess when the physics gets more advanced the solutions in games will become a lot more hacky, but yeah it is really cool these are basically one to one implementations. If you want to you should give it a try, especially in the end with collisions it’s gotta be a lot of fun to see the results. But I still have to do a bit of work until that‘s done

[@tempname8263]

Yeah, you don't need an entire matrix though. One (diagonal) vector is enough. Inertia tensor is just meant to resize momentum into a velocity, and that's all it does. Insanely simple idea.

[@A_Random_Ghost]

Nice, very nice.

[@blackedoutk]

Thanks :)

[@Tomyb15]

Amazing video. Really well done and I loved how you approached the topic. It was also really funny at times! I was laughing when the tension over baiting an explanation on quaternions over matrices scene happened, and I was genuinely lol'ing when the terminal started showing random strings of characters as the guy in the clip smashed the keyboard. If there's any criticism I can make is that you sound a bit monotone. Many of the jokes would have been 10x better with some small comedic performance. Dude, you are funny and great! I know you can do a version with slightly more energy even if just for the joke parts. Please keep this content up! Subscribed.

[@blackedoutk]

love your comment, made me happy thank you :)) I agree with the monotone bit but it's hard for me to change that. sometimes I think I already bore people when I'm just telling them stories but it's the worst when I'm reading out loud 💀 plus this is english and I have to calm myself because somehow I get stressed when recording the script and then I sound like in the first or second epsiode. I'll see if I can do something, though doubt it. but definitely a fair point, appreciate it

[@弘睿甫]

Epic video. The most relatable and useful one so far

[@blackedoutk]

That‘s great, thanks a lot

[@blackedoutk]

did you delete your reply or was it some yt shenanigans?

[@弘睿甫]

@@blackedoutk I did. I wasn't in my best state of mind and thought i was being cringe so i deleted it lol. but my point still holds, epic channel bro

[@blackedoutk]

@@弘睿甫 no worries I am the same. part of why I am so slow to answer is because I overthink what to say. sometimes I respond faster and cringe a month later lol. I thought your comment was funny

[@Temple_Cloud]

Love it! Thanks! Keep them coming!

[@blackedoutk]

Great to hear, thank you

[@ladyaliciaherrera3437]

Excellent video!

[@blackedoutk]

Thank you! :)

[@stick-Iink]

Really nice video!

[@blackedoutk]

Thank you!

[@niklaswagner6191]

Very nice video on such an interesting topic. You've actually explained the spin instead of the angular momentum since you were using r and not x. In your notation it would be the angular momentum L = x x p and the spin N = r x p (with integrals). The problem you've faced with the referential density also holds for the integration limits. They would be also dependant on the actual rotation when you are using the inertial basis as coordinate representation. I made the observation, that many people get problems with rotations and transfomration matrices. The vectors r, r-~ and r-v all refere to the same vector in space, namely the vector from CM to the particle, but only represented in other coordinate systems. There isn't something rotated, it's just an projection to other basis vectors. You could try to store the Omega and N in body fixed coordinates, how it is often done in mechanical simulations. I guess that could save some matrix multiplications. Thanks for the video, I like to see you making progress. I liked the explanation to get the angular velocity with the rescaling, made it more intuitive what's inside the inertia tensor.

[@blackedoutk]

Thank you :) Definitely interesting, I didn't know there was a differentiation between the spin angular momentum and the orbital angular momentum. But I don't think it's necessarily wrong to still call it angular momentum, just a matter of which origin point is used. How would the density be dependent on the rotation in an inertia basis? I am not sure I understand. Also, while r, r-~and r-v refer to the same vectors in local coordinates, I am not sure this is the case for global coordinates. Maybe you have another intuition than I have? I thought about storing the angular momentum in local coordinates, this might be a good idea. Though I am not exactly sure because then you might have to rotate the torque vector. What would be the benefit of storing omega in local coordinates? I really appreciate your remarks, thanks

[@niklaswagner6191]

​@@blackedoutk Thanks for your reply. I wasn't conscious of these english expressions for spin angular momentum and orbital momentum. For kinematics, you have to use the same reference point for induced torques, which is anyway convinient to take the center of mass for that. Something that really improved my unterstanding of physics and especially mechanics, was to think of vectors as arrows in the space instead of putting three number on top of each other. Mechanics works as well with these arrows, but we need number to describe these arrows. For that we can use a inertial fixed basis or any other, like body fixed. Then, r, r-tilde and r-v describe the same arrow in space, but with other numbers. After rewatching your the sequence I think I got it better. Your density function (rho) takes the vector in global coordinates and rho-tilde takes it in body fixed coordinates, which would be the only possibility to give a proper definition of the function. Lastly, I have never thought about integrating the angular momentum instead of the angular velocity. This could simplify many things and removes some exhausting derivatives and materix multiplications. I think expressing the torque vector in local coordinates could be already cheaper. Force elements are mostly mounted body fixed, so for rotaional springs/dampers, their force contributions are already in local coordinates. For induced torques by a globally given force, you either have to take force into local coordaintes or calculate r in global coordinates. Would be interesting to see some benchmark experiments with both approaches and whether you face any other problems like in time integration. Anyways, I am curious see the progress you're making :)

[@blackedoutk]

​@@niklaswagner6191 Such benchmarks would definitely be interesting. Though later I will probably stick to storing and integrating the angular velocity anyway, because that's how the XPBD paper does it. Your understanding of the density functions is how I imagined it, so the definition would then be rho(R, r) = rho-tilde(R^T*r). I try to think of vectors as arrows in space too. And even matrices as multiple arrows if applicable. I believe understanding these concepts is much easier with a visual intuition and find it quite unsatisfying when people introduce some theory (for example the inertia tensor) without providing an intuition for it. In my opinion, there should be some amount of focus on teaching that as well

[@jaborl]

At 26:15, you said matrix multiplication is commutative but it is actually associative.

[@blackedoutk]

Thanks for pointing that out. I meant to say associative. Matrix multiplication is not commutative in general.

[@Renegen1]

great intro!

[@xydez]

bloody incredible, I love to see a mix of math and implementing it into code

[@blackedoutk]

Awesome, thank you 🙏

[@MrBomberman11]

This is my new favorite video on KZbin

[@blackedoutk]

Woah, really cool. I'm happy you like the video so much :)

[@MrBomberman11]

@@blackedoutk I wrote a game engine for a uni project and the physics transforms really tripped me up. But this video and the follow up one on inertia made everything so much clearer. Do you have another dev log planned soon for fixing the determinant bug and other juicy stuff?

[@blackedoutk]

@@MrBomberman11 Planned yes, but without date. So next video would be intertia tensor decomposition and then the next I wanted to fix the determinant using quaternions

[@iizvullok]

0:20 I am glad to learn that i am not the only one who makes up all those excuses.

[@blackedoutk]

haha, I mean it's not even wrong is it?

[@absence9443]

Excellent! Especially because the video requires the exact amount of math I'm familiar with :)

[@chillyvanilly6352]

ABSOLUTELY AMAZING VID!!! And yeah...the intro couldn't be more true * sigh * But there is still kinda a lot of additional background even to such an amazing vid as this to grasp thing, you know...for us mere mortals that are not as proficient with math yet as you mate may be xD One of those things is "the 3 pressure terms" referred to as "stresses" acting on an object as physics seem to tell us (forgot whos law that was exactly, read it somewhere in a NASA/TM paper). FYI: spotted typo at 30:45, "square" ✅ instead of "sqaure" ❌ P.S.: danke dir VIELMALS fuer die tolle Erklaerung und die Visualisierung der Dinge/Konzepte! (wollte diesen Kommentar eigentlich as "Super Thanks" mit ner donation senden, hast aber anscheinend nicht enabled fuer dein Vid :/ )

[@billy.n2813]

Thank you so much for this

[@conniichan]

I’ve literally been looking for a video like this! I’ve always wanted to program physics simulations. Hope I have the sufficient math knowledge in the next upcoming semesters :)

[@blackedoutk]

Nice! Programming physics simulations is really fun (if it works:D), unfortunately the math can be very challenging. You studying computer science?

[@conniichan]

@@blackedoutk I'm studying computer engineering and have an interest in physics :D

[@blackedoutk]

@@conniichan Ohh the lower level computer science, right? Is it a bit of physics aswell? Porbably more circuit physics

[@io-mi]

Here is code for calculating mass of any triangular mesh with uniform density. def get_volume(indices:list, vertices:list) -> float: volume = 0.0 for i,j,k in indices: a = vertices[i] b = vertices[j] c = vertices[k] e0y = b[1] - a[1] e0z = b[2] - a[2] e1y = c[1] - a[1] e1z = c[2] - a[2] volume += (e0y * e1z - e0z * e1y) * (a[0] + b[0] + c[0]) volume /= 6.0 return volume def get_mass(indices:list, vertices:list, density:float) -> float: return density * get_volume(indices, vertices) If you need Inertia tensor as well, it could be calculated at the same time with some changes. Note: triangle winding must be CCW.

[@blackedoutk]

How did you come up with this? I tried to make sense of it but couldn't.

[@io-mi]

@@blackedoutk it's basically a volume integral reduced to a surface integral using divergence theorem.

[@blackedoutk]

@@io-mi Ah nice! That reminds me of the paper "Fast and Accurate Computation of Polyhedral Mass Properties" by Brian Mirtich where he uses the divergence theorem to compute the inertia tensor. It was actually the first method I tried to use, but found it to be a bit too complicated 😅 Although the volume computation you posted looks quite cheap. I wonder how it performs in contrast to the scalar triple product version

[@io-mi]

@@blackedoutk By the scalar triple product version you mean the one where you calculate volume of each tetrahedron? What's cool about this version is with some changes it could calculate the intertia tensor at the same time. IIRC it's from a paper "Polyhedral Mass Properties (Revisited)" by David Eberly.

[@blackedoutk]

@@io-mi Yeah the volume for each tetrahedron one. Yours has 2 subtractions more but only 3 multiplies instead of 9. But it's probably more about how simd compatible they are. You can actually combine the inertia combinutation with the volume for each tetrahedron version as well. In fact, that's what I did. Hopefully I can talk about that in the next episode.

[@Calculsus]

Please make a video (if you haven’t already) implementing the quaternion approach. That would be really fun to watch. Also great video, really enjoyed it.

[@blackedoutk]

Thank you :) my plan was to cover the quaternion implementation in episode 8

[@azmah1999]

Funny how at 6:15, you say you think of momentum as the "amount of movement power", since in French momentum translates to "quantité de mouvement", which literally translates to "amount of movement". So you actually were right all along haha. Great video !

[@blackedoutk]

Haha, thank you. That's so cool. In german it's very confusing unfortunately because momentum translates to "Impuls" but english impulse has another meaning and also the word "Drehmoment" which has "Moment" in it means torque

[@saiyamjain772]

bro i didnt see your video but you are living my dream ! , i wont see your video until i try to do this myself and then see how far i go !!!!!!!!!

[@blackedoutk]

nice, let us know how far you've come

[@slavicradko9846]

This video is great, thank you very much! I will certainly try it out myself in my code.

[@blackedoutk]

thanks:) yes definitely try it out. working on a game or just for fun?

[@slavicradko9846]

@@blackedoutk I have a pet project in C from scratch, where I learn things. Based on handmade hero, but with my own twist. I have an idea for a game that is reachable for my abilities, I think, but it's still a huge amount of work, and it's really hard to combine with a full time job.

[@blackedoutk]

@@slavicradko9846 oh cool, I also watched handmade hero, but only the first 200 episodes or so. it's really great. did you watch the whole thing? I can imagine the difficulty when having a full time job. I don't have one but still struggle with time management :/

[@slavicradko9846]

@@blackedoutk no, I did not, I watched around 100 and then selectively some of the later. It was enough to get me a well paid job though, so I greatly appreciate Casey for his efforts 🙏

[@AMax364]

Transposes R equal to inverted R if R is orthogonal matrix

[@blackedoutk]

Did I forget to mention that?

[@AMax364]

I recommend you to search about "matrice d inertie " and "tenseur dynamique" this is everything you explain but in very simple form

[@blackedoutk]

I doubt this is everything I explained and also I don‘t understand french

[@AMax364]

is method we use for simplify the calculation

[@cbbbbbbbbbbbb]

Oh man, I was so disappointed at 17:30. I thought for sure you were going to use quaternions. That's what I've been spending time learning and using recently so I was excited to see if you would take it there. Great video nonetheless!

[@blackedoutk]

I totally understand, quaternions are what you would want to use, but I think it's easier to understand when starting with rotation matrices. I want to introduce quaternions in a future episode. Happy you still like the video though, thanks

[@zvisger]

😮Ah I see. Yes, I made it through the video and understand 100%. This is just some nice, easy math.

[@dumdum7099]

Wait this is a very valuable resource. Damn.

[@blackedoutk]

Glad you think so :D

[@sargates]

I was really hoping you would use quaternions in this video, there aren't enough good videos about quaternions. Great video none the less

[@blackedoutk]

Oh. Sorry :/ What specifically would you want a video about quaternions to cover? I think the hardest part about them is getting a good visual intuition, which I currently don't have myself. I am glad you like the video despite missing quaternions

[@Tannz0rz]

@@blackedoutk All of these computations can be performed using geometric algebra, which generalizes quaternions/rotors. I would encourage you to look up “May The Forque Be With You” by Leo Dorst & Steven De Keninck.

[@blackedoutk]

​@@Tannz0rz Ah, I've glossed over their website before, but it looks so complicated. What would be the benefit of understanding it?

[@sargates]

​@@blackedoutk For visualization there's the 3B1B video (which I'm sure you've already seen and if not give it a watch), but it *is* something that's 4D so there's never gonna be a perfect way to understand them visually (maybe I just tell myself that because I don't). What I'd like to see is a somewhat in-depth explanation on the implementation of how you might set them up and any of the math required to do that. I'm sure that if you ever touch on them in a future video it'll be great.

[@blackedoutk]

@@sargates :D I have seen the 3B1B video at least partially, but it was quite some time ago, so I don't remember much. Might have to watch it again. Though implementing quaternions is doable without too many headaches I think, unlike understanding their 4D intuition. But maybe the geometric algebra guy in this thread has a point too. I just watched this video here kzbin.info/www/bejne/f5XPp2tpeN2DY5o and it sounds quite promising. Perhaps there is a better way to understand quaternions. It's just that there are a lot of other new concepts in geometric algebra as well.

[@jb14_99]

Awesome video. Very cool

[@blackedoutk]

Thank you, happy you like it :)

[@Waffle4569]

C++ - "It works on my machine" the language

[@blackedoutk]

is that an insult 😤 what's your fav language

[@Waffle4569]

​@@blackedoutk Referring to the build system that even you are having horrible issues with. And, Rust

[@blackedoutk]

@@Waffle4569 Ah I see. But I think that's more of a problem with the build tools rather than the language itself. Maybe it's because I have never worked on big enterprise software, but I really don't understand the need for all of the complexity in build tools. For small to medium sized projects, using shell scripts is definitely superior imo. Also because you say "even you", I don't think I am a good reference for comparison when it comes to building projects, I'm horrible at that. I tried rust recently because I have to use it for a project and I get really angry about the borrow checker all the time lol, but I know that's probably a skill issue

[@Waffle4569]

@@blackedoutk I think it stems from the simplicity of the old days when compiling was just calling an exe with arguments, but its horrendously overgrown now. The build system was never intended for more than one person's personal computer, absurd amounts of build config are defined by command line variables and dependency directories that might not be present. Not to mention there is no real compiler standardization, so even the compiler becomes a dependency that might not be listed. Pretty much every language after C++ realized they need to define how the build system and dependencies should work. Rust has a very steep learning curve, but ironically it has the easiest dependency/build system I have ever used.

[@muhdiversity7409]

CMake is cancer of the highest order. It is the PHP of build systems. And that is an insult to PHP.

[@netwonderer]

Really, asides the fact that your video is amazing. Your deep voice makes me wanna sleep... 😂

[@0dWHOHWb0]

23:47 You forgot to tell us what tmn and tvn are

[@blackedoutk]

They are template structs for vectors (tvn) and matrices (tmn) where the parameter T is the scalar type and U is the number of elements per row or column. The data is stored in an array named E, hence why in code you always see .E[] instead of x, y and z for example. To make these easier to use I then typedefed them like this: typedef tvn v3; and with a prefix d for 64 bit floats.

[@0dWHOHWb0]

@@blackedoutk Ah, okay -- thanks!

[@apmcx]

Excellent video

[@blackedoutk]

Thanks, glad you like it :)

[@tempname8263]

tl;dw Let me try to make it shorter. Body has linear and angular momentum. Divide linear momentum by mass to get linear velocity. Transform angular momentum into local frame of reference, divide each component by one of 3 angular masses, and transform it back into global reference frame to get angular velocity (which doesn't always ends up aligned with angular momentum, creating interesting dynamics). Or compose it all in it's entirety into a single momentum->velocity transformation, if you so desire, which ends up as a symmetric matrix - matrix that resizes space relatively to some orthonormal basis. Now move your position by vel*time and rotate your rotation by angVel*time. The tricky part here is actually calculating next rotation, since it is represented *either* with geometric algebra rotors and bivectors, or using matrices. (And no, euler angles don't count, they are just funny useless parametrization, that can't do anything on it's own) Anyway, long story short(er): Angular velocity (or momentum) is a bivector - a quantity, where each component is associated with 2 coordinates. Kinda non-diagonal elements in a matrix. But here, instead of linear mapping, they represent oriented quantity from one axis to another one - for example "+5xy" shows rotation around "z" axis. Easiest way to turn them into actual rotation is by viewing them through the lense of VGA. In it, vector is used to represent reflection around a plane, and so a composition of two of them gives us a rotative transformation twice that of angle between them. This is called a rotor (in 2d only it's a complex number, in 3d it's quaternion). Composition rules are pretty simple (read left to right): "x*x = 1", "x*y = xy". So "(1x + 0y) * (0x + 1y) = 1xy". "1xy" in this case is just a bivector, since vectors were orthogonal to each other, and it represents a +180 degree rotation. But reflections generally don't commute, so composing them backwards will give us opposite rotation "y*x = -1xy", which is -180 degree rotation around z (in 3d). Now, these bivectors can be composition-exponentiated, which gives us a rotor (in 2d and 3d you can implement exponentiation using very simple trig, and for other dimensions there is a more general formula). In other words, we just turned rotation direction into a rotation. Compose it with our original rotation, and you're good to go. In any amount of dimensions, at that. Have fun rotating hypercubes! Now, in order to render vertices and whatnot, just transform your vector represented as a reflection-operation into a global frame. How? The same exact way we did with local angular mass (which is just an operation converting local momentum into local velocity) and rotation. "[madeupArgument]*reverse(rotation)*reflection*rotation" "[madeupArgument]*reflectionOutsideRotation". Anyway, I'll go into more detail and fill in the gaps, if anyone ever reads this and wants to know more. Could as well touch on PGA rigidbody dynamics too, since they are quite elegant and interesting, although their math is a bit too big-brained and not as general purpose, as concepts I discussed.

[@tempname8263]

tl;dr

[@blackedoutk]

I think covering this in depth needs a bit of time, hence the length of the video. And while I appreciate you trying to explain rotors and bivectors, I don't understand this purely in a short comment. Why don't you make a video about it?

[@tempname8263]

@@blackedoutk Uhhh.. Yeah, making videos is not mine kind of thing. I'd probably just write an article series in couple years, after my current projects are over

[@tempname8263]

@@blackedoutk Just check out materials created by bivector community, to fill in the gaps. My goal was just to give a rough intuition, which those materials on their own do not fully provide. They are more focused on algebraic aspect, and working with transformations as if they're objects (by talking about transformation's invarient subspaces).

[@linuxgaminginfullhd60fps10]

As you mentioned one way to achieve numerical stability is to apply corrections. Another way you mentioned would be to use quaternions. Those are popular solutions to the problem, however there are some others. There is no research required, all the knowledge is already there, yet it is kinda distributed between different disciplines... You might want to familiarize yourself with "second quantization". All the constrains of the physics model can be embedded in it and in your case that would be a discrete physics model, unlike the continuous models we usually use to describe real world. Thus all interactions of the objects in your engine can be formulated in terms of some integer values, which do not have all those problematic rounding errors, which later result in numerical instability. You probably realize how difficult quantum, mechanics is in terms of those complex probability density functions. And then you multiply one very complicated unmeasurable complex function with another one very complicated unmeasurable complex function, integrate it and whaaaat?!?! the result is just a integer number. So instead of getting all the numerical problems for the problem formulated in terms of hard to simulate continuous functions, which one might try to approximate, you could just build the physics based on discrete model from ground up. So the physics state would be consistent and exact at every single update of the simulation, while approximation would only be made when the frame is rendered.

[@lupino652]

Quaternions are used in the software of arm robots, because it doesnt have singularity points like using euler angles or Rodriguez fórmula. After that you are stating facts of numerical analysis and optimization, the best way to simulate rigid body physics is using tensors like prior, back, kullback, etc it is solid computational mechanics. That with CSD and CFD to model plasticity. Quantum mechanocs has nothing to do with rigid body mechanics

[@linuxgaminginfullhd60fps10]

@@lupino652 My point was not about using quantum mechanics. My point was about using native systems symmetry that would allow exact computations in a simulation with limited precision. Second quantization is just an example how physicists made it work for quantum mechanics. I am not against using quaternions, or matrices, or any other very useful construct. You would still have to use them for derivation of proper formulas. I am just saying that the errors caused by rounding can be entirely excluded from the simulation with a slightly different representation of internal state(using integer quantities like amount of action, tick number, occupation number, energy level number).

[@blackedoutk]

I want to use quaternions in the future to replace the rotation matrices. Really have no clue about quantum mechanics. If you could just use a different model with integers and have no numerical errors, why is it not done everywhere? Too expensive?

[@madhavgoyal6093]

Great explanation

[@blackedoutk]

Thanks :)

[@gedaliakoehler6992]

Very good video. Nice job.

[@blackedoutk]

Thank you

[@TheyCallMeApplePie]

I have no idea what is happening. I barely understand the math, but something inside me made me click the video and subscribe

[@blackedoutk]

I like that:D I mean, not the part that you don't understand the math. Is there something I can do to fix that?

[@pugglez4798]

This is great stuff, I subscribed. Looking forward to future videos incorporating forces and accelerations

[@blackedoutk]

Nice, thank you

[@user-3bs8jd83js]

14:30 Why not just use trapeziums to calculate the approx. area? You can calculate the next v 1 iteration before, store it, calculate the correct trapezium area, and then the next iteration you already have the current v.

[@blackedoutk]

Yes that would be better, however keep in mind that you need the right end value to do that, which means it's an implicit method and thus way harder than explicit. It's a bit difficult to see with the linear velocity example because the velocity does not depend on the position, so you can just compute it here. But try to write down the same thing for the angular velocity. This one depends on the current orientation. So to get the exact right end value you need the angular velocity there. However this velocity depends on the orientation at that point in time. It's a cyclic dependency. Does that make sense? There are explicit methods that do kind of what you suggested for simple functions, like the midpoint method which is second order explicit. But of course the formula looks more complicated so I wanted to stick with something simple here.

[@Hector-bj3ls]

So far I've found Verlet to be better than Euler. It's more stable. I've been building a cloth simulation, and with Euler it explodes way too often. Verlet doesn't use an explicit velocity, it calculates it from the previous position: vel = pos - prev prev = pos pos += vel + acc * dt * dt

[@blackedoutk]

Yes there are definitely problems where it is more stable. I think it is closely related to the semi implicit Euler method

[@tharteon1866]

The boost part cranked me up, you gotta hate how boost is conserved between C++ projects, and how hard is it to install always

[@blackedoutk]

haha yes :( though I'm generally bad at using build tools

[@renuk8560]

7:13 "Dot notation" is actually called newton's notation.

[@blackedoutk]

Ah interesting. Sometimes I just use non technical terms to make the stuff I say more relatable or easier to understand. Though admittedly I didn't know it was called newton's notation

[@renuk8560]

@@blackedoutk btw bro, why did you stopped uploading?😭 Your content was so fire 🔥🔥🔥

[@blackedoutk]

@@renuk8560 university :/ Right now if I could I would be programming all day haha

[@Izz4cc]

Thank you for this video! I've always enjoyed math and physics applied in programming. Btw at 26:14 , matrix multiplication is NOT commutative, (AB≠BA (usually)), rather, its associative (A(BC) =(AB)C)

[@blackedoutk]

me too, I wish there were more videos that guide you through the details. and you're right, that's a mistake in the video. luckily I just mixed up the two words so the math is still correct. I had put this in the pinned comment, but maybe it's too long already 💀

[@Izz4cc]

@@blackedoutk oh sorry my bad I didn't see that comment

[@blackedoutk]

@@Izz4cc no worries

[@korigamik]

I am loving all of the explanations! Can you share the source code of the animations in the video? How do you do the voiceovers? Will you share your right body simulator code as well?

[@blackedoutk]

that's great, thank you. sharing the 3d animations is difficult because they are part of the game, so for anyone to understand the code I would have to share parts of the game code aswell. the manim animations I could share but I'm not sure I want to, what would you want to do with them? for the voice overs I just record me reading the script and then fiddle the stuff together in DaVinci Resolve. I will share the rigid body code eventually in the library I talk about in the episode 6, but the game code for now I don't want to share. my goal is to explain the important parts, so that viewers can implement it themselves or at least know where to look for

[@senkrouf]

I was thinkking of an energy based physic system, I think using vector laws is a bad idea for videogame because eventually everything expldoes. A scalar field physic systems with the forced constraint that energy should increase could do the trick, the biggest problem is dealing with edge cases of the discrete time.

[@peppescala4113]

I have a problem at 09:44. As a theoretical physics graduate student I'm used to C and Python so I understand the idea behind your struct, but what kind of libraries are you using? I've never seen "dv3" as a datatype to define 3D vectors as well as the datatype "mesh''. Can you provide some info? Beautiful video by the way!

[@blackedoutk]

Thank you, glad you like the visuals! The data structures you're seeing are part of the game and not from a library. I talk a little bit about the general vector type in my first episode, but really they are just structs that hold the coordinate data in an array. Then I wrote all of the functions needed to work with them, like operators, dot products and so on. The mesh struct I don't think I have mentioned in detail in any episode. It holds all of the data for a model, like vertex positions, normals, tex coords, colors, face indices and textures, and the handles that are needed for the OpenGL rendering, meaning vertex array object, vertex buffer object and texture handles. It also contains some less relevant stuff like the models name or edge information which lets me draw a wireframe model without using wireframe mode. Lastly it contains the original center of mass and decomposed intertia tensor of the model as it was exported from blender for example. Though this I plan on explaining in the next episode. I updated the description with the libraries the game uses in its state of the video. Let me know in case you have further questions about this

[@Shikinoe93]

What a coincidence! I've also been working on implementing some rigidbody physics. I am currently trying to implement angular motion with a constant angular velocity, just for simplicity. I have a question: I have tried updating the rotation matrix as you did in the video: Cuboid->R += DeltaTime*Cross(Omega)*R (with a constant omega vector) However, when I do that, my rotation matrix is no longer a rotation matrix after a few thousand simulation steps (The determinant explodes into values like 200). I'm using a DeltaTime of 0.01, but I have the same problem with a DeltaTime of 0.001. Am I missing something? Do you normalize at some point? Thanks for the video btw

[@blackedoutk]

Cool! The problem you are facing is completely normal, I talk very briefly about this at the end of the video. But to add to that, you can even make intuitive sense of it. What you are trying to do by adding Cross(Omega)*R to R itself is move the axes of R along their respective circular paths. As explained in the video, Cross(Omega)*R contains the velocities of each axis vector of R, which are tangent to the circles. Regardless of how small your step is, if you move some epsilon > 0 along the tangent line of a circle, you will always leave the circle. Resulting in a larger radius than previously. The axes increase in length. You can fix this by reorthogonalizing R after each simulation step if you wanted to continue to work with a rotation matrix. Another, probably better fix would be to use quaternions instead. I didn't do any of that in the video, for some of the footage I just used an even smaller time step 😅

[@Shikinoe93]

@@blackedoutk Thanks for the answer. Yeah, I figured out that it was normal but I didn't think the drift would happen so fast especially because it seemed to work well in your video 😅. I implemented a solution using quaternion too but can't seem to figure out what to do with the derivative of the quaternion after computing it. Simply adding the derivative x DeltaTime doesn't seem to give any good results...

[@blackedoutk]

@@Shikinoe93 Probably because the captures weren't very long. I think I used 1ms for most of it and only 0.01 ms for the "phone" and the screw thing. Adding the derivative x DeltaTime should work. Though similar to how only orthogonal matrices (with a determinant of 1) describe rotations, only quaternions of unit length describe rotations. So you might want to normalize your quaternion after adding its delta value.

[@Shikinoe93]

@@blackedoutk I'm getting much more stable results now with quaternions. The previous implementation with rotation matrices would explode depending on the DeltaTime. Still not getting to the precision I need though, I don't know what more I could do to improve the accuracy of the simulation. Even increasing the deltatime doesn't seem to change much...

[@blackedoutk]

@@Shikinoe93 Hmm, I would need to know more about your implementation to see what might be wrong. Are you using double precision?

[@pushqrdx]

Great video, I have a question though, didn't you mention in one of the soft body videos that you can use a constraint to essentially have the effect of rigid body?

[@blackedoutk]

Thanks :) I don't think this would work using a single constraint, but with multiple it might be somewhat possible, like for the "rigid" cube in episode 1. Are you referring to that? There are a few problems when using multiple constraints to make an object rigid like in previous episodes. First, the mass properties aren't exactly correct, because you are dealing with point masses instead of a contiuous body. If you didn't tetrahedralize your mesh, the mass would even only lie on the surface of the model. Secondly, it is probably inaccurate and or very expensive, depending on the complexity of the model and how many iterations you perform. In comparison, simulating the dynamics of a rigid body like in this episode is almost free and independent of the model's complexity (assuming the initial inertia tensor is given). As an example, in the previous episode, if the solver was perfect, the tire wouldn't have been soft at all, because I set its inverse stiffness to zero. But since I only performed a limited number of iterations, which were already pretty expensive, the tire appeared to be soft.

[@pushqrdx]

@@blackedoutk I see, and yes, I was referring to the cube, great explanation. Thank you very much and keep up the videos I really enjoyed them.

[@thebumblecrag61]

Yes!

[@blackedoutk]

🎉

[@minma02262]

You are my god.

[@blackedoutk]

woah not so fast, where are my prayers?

[@TheNitroPython]

The first minute perfectly describes my experience with this lol

[@amperev2807]

Yo this is just what ive been looking for! Could I ask what program you used to animate these slides?

[@blackedoutk]

Nice! So basically the video is a collection of images, screen recordings, manim animations for the equations and custom animations that I made in my game itself. All of these I then put into DaVinci Resolve with my voice and added some further texts, emojis and stuff. The custom animations are also screen recordings because I didn't want to spent even more time trying to export the rendered images as a video. You can see this even at 32:49 where I struggled to remove the background from the object when its duplicate slides over 😅 But they are in game renders. Was a bit of a pain at first because I had to write the internal animation framework while creating the final animations for consistent playback, mouse handling and stuff. I hope this answers your question. At first I was thinking about using blender for the custom animations but I wanted to use my physics code in them

[@echovictordcsworld]

To ensure the determinant of the rotation matrix remains at 1.00000 can you not simply normalize the x y and z components before applying it to the mesh? Loving the videos, i am trying to create my own N-body gravity simulation in C++/OpenGL, and have been trying to wrap my head around rotation natrices and this video helped perfectly.

[@blackedoutk]

That's great to hear :) I haven't tried anything to fix the rotation matrix so far, but I think that while just normalizing the axes might help, it doesn't guarantee orthogonality. So you also need to ensure that all axes are perpendicular to each other. For that you could use cross products (normalized) or something like Gram-Schmidt. But I don't know which approach works best.

[@richardbloemenkamp8532]

@@blackedoutk The book Game Programming Gems 1 for which you find a pdf online has a chapter 2.2 on "Integrating the Equations of Rigid Body motion" by Miguel Gomez on this subject. This chapter proposes a different integration variable "q" and also some references because it claims that the rotation-matrix method introduces error and requires regular reorthogonalization. BTW a lot of this rigid body math is also present in university level courses on mechanics, robotics and aerospace engineering.

[@blackedoutk]

@@richardbloemenkamp8532 Nice, thanks for sharing the resource. They use a unit quaternion q instead of a rotation matrix to avoid having to do a reorthogonalization. The quaternion just needs to be normalized after the update, which is easier. In the future I want to use quaternions too, but here I wanted to start with the basics. I noticed that too, that rigid body math is part of quite a few university lectures. Currently I am attending a robotics lecture and a lot of the stuff I researched for this video helped me to understand some of the topics faster. They also presented algorithms that build on top of these basics, like for chains of rigid bodies with joints (robots).

[@tempname8263]

Best way is not to use rotation matrices. Just use rotors. Uhhh, most people aren't educated about them, though, right? Yeah, they call them quaternions/complex numbers :d

[@samsaraAI2025]

Thanks! I understand more less how L is conserved at minute: 38:30, but how do you apply an external force to the L. I mean the combination of the L conservation and also some forces producing torques. Thanks!

[@blackedoutk]

Similar to the linear motion where force is the derivatve of the linear momentum p, torque is the derivative of the angular momentum L. So in both cases, applying forces or torques for a certain time at the center of mass can be accounted for by integrating both of these and adding them to the respective momentum variables. If you have an external force that is not applied to the center of mass, you have to apply it to the center of mass and separately compute its resulting torque by using Torque = Cross(point where the force is applied, Force) and apply it to the angular momentum too. Does this help? I would touch on this in future episodes, but idk when I will do those

[@mr.pumpkinn]

10:25 What is bold *x* exactly? Is it like position? Position of what?

[@blackedoutk]

So at the timestamp you've given bold x would be the column vector of x, y and z. And because we integrate over the domain Omega (the cube) bold x is the position of every point inside of Omega

[@mr.pumpkinn]

@@blackedoutk ok thanks

[@NatoSkato]

Wow youtube compression is very good. Downloaded this video because it is great and found out it's only 110 megabytes!

[@monx]

i laughed out loud at 17:20

[@blackedoutk]

:D

[@notu483]

Perhaps some parts weren’t too rigorous (such as the part where you exchange the derivative and integral operator, which is actually a specific case of the Leibniz integration rule where the region is constant), but it is very valuable nonetheless. Thank you! ❤

[@blackedoutk]

Interesting, I should do some more checking instead of relying on intuitions too much maybe. So the explanation for the factoring out isn't really correct, right? I looked at the Leibnitz integration rule for higher dimensions on Wikipedia (it also says Reynolds transport theorem on there) and noticed that I probably should have been more specific what the x really is because it doesn't distinguish between the x of the domain of integration and the resulting x in world coordinates. But when I do that (e.g. use w(x, t) instead of x(t)) then I think the "factoring out" step still works because the velocity of the domain's boundary is zero (since it's static) and the partial derivative equals the total derivative again because x doesn't change over time. Really appreciate your remark thank you, and I'm also happy you still think it's valuable. Now I'm curious, are you a mathematician?

[@manfredbogner9799]

Sehr gut

[@blackedoutk]

Freut mich

[@dixztube]

Good video

[@blackedoutk]

Thanks:)

[@aragon5956]

What course do you recommend to learn that? whether it’s a book or an online course, with the required topics in math or physics?

[@blackedoutk]

hm it's hard for me to answer because there isn't one specific book I read or course I watched, rather it's small bits from everywhere. one commenter said the first half of a transcendental calculus book should be sufficient for anybody looking to learn. but I can't verify this. definitely a bit of calculus, linear algebra, numerics (ode and linear algebra) and a mix of calculus and linear algebra, so multidimensional calculus, where you learn about the jacobian matrix. maybe the "essence of" series by 3blue1brown is a good start?

[@eliaspierre6303]

I'm having some trouble implementing this into a blender simulation with simulation nodes. I've been trying for about a month, I keep coming back to this video as it is describes a simulation with assumptions very similar to mine such as using only point masses, constant density, etc. I was wondering if I could send an annotated photo of my simulation setup to you through email. I know you're using c++ but the node I'm using in blender are quite simple so I'm sure you'd be able to understand them. If not no worries though. Either way thanks for this video it explains it very concisely!

[@blackedoutk]

Unfortunately I don't have much time at the moment, but you can still send it to the email that is listed in the channel details/info. In a month or so I might be able to take a look at it. Apart from that you can also join my discord server and post it on there if it's not something that you wish to be kept private. Maybe somebody over there has an idea why it's not working

[@maz3808]

Facinating video. Are you interested in computational EM and opics? or just physics engine and rigid body motion?

[@blackedoutk]

thanks. currently more interested in rigid and soft body physics. but I'm not opposed to other simulation topics, just already so many things to learn. do you know more about these other topics? or do you wish to watch videos about them?:D

[@maz3808]

@@blackedoutk Yes I know more about these topics and I wish to watch videos about them too. There are channels that do something similar but I thought why not ask you about your interest to understand you area of expertise.

[@dimitrisgkofas7787]

I have done something like this with joints for objects in java do you want the code to see if it is ok?

[@blackedoutk]

I can take a look if you want, but I am actually learning these things roughly at the same time as I am making the videos about it. So I haven't implemented joints myself and I am not sure if I can validate your implementation

[@philtoa334]

Nice.

[@blackedoutk]

glad you like it thank you

[@scoutgaming737]

26:20 Why is the vector on the left and matrix on the right I'm scared

[@pavelyankouski4913]

The scene reminds Unreal Engine room ^^

[@rutalorp4777]

14:00 yooo I used a pretty basic numerical approximation like this in my double pendulum simulations so that's why it blew up really fast

[@blackedoutk]

haha yes, given enough time this will happen unfortunately