To give some context, this is only part one in a series of blog posts I plan on writing about rigid body physics.
The post is aimed at people like myself, who aren't game devs and don't necessarily have a strong math background. Which is why I spend so much time explaining concepts that would appear almost trivial to someone who has experience in this area.
Feedback: Your introduction "From Mario bouncing off a Goomba..." might be a bit misleading IMO because most games like the classic Super Mario titles on NES and SNES do not require and did not use most of these calculations.
Game development beginners often have the wrong impression that they need rigid body collision calculations or a 2D physics engine like Box2D to handle collisions. That's true if you want to make a game like Pool or something with collapsing stacks of crates like Angry Birds.
But for a 2D platformer you only need to detect collisions by comparing (axis-aligned) rectangles and to handle collisions by changing the moving character's X and Y coordinates (to undo an overlap) or setting the character's Y velocity (after using the jump button, or after landing on a Goomba's head).
This also makes it easier for the developer to finetune exactly how moving the character should feel like. (This includes inertia, but this inertia is usually not physically realistic.) Trying to use realistic physics as a gamedev beginner can easily lead to floaty and unsatisfying movement.
Very nice. The section "A word about math" really is important. I am admittedly not good at math, but I was able to build a VERY (extremely, even) rudimentary physics simulation many years ago by really really simplifying the math concepts. I was essentially iterating on the building blocks, points, lines, etc. With a lot of small steps and a lot of visual debugging lines, it ended up extremely janky and slow, but it did kinda work!
Great article and very fun to read, as someone who also doesn't have a strong math background, so thank you for explaining these "trivial" concepts :)
Are you planning to read/explain through XPBD (Extended Position Based Dynamics - http://mmacklin.com/xpbd.pdf) as well in future posts? The concept seems to be gaining traction and I've used it with Bevy (via https://github.com/Jondolf/bevy_xpbd) with big success so far, seems more stable than the usual approach.
Most of the physics-based example are written in WebGL using three.js. Using three.js is a little bit overkill for this because all the examples are in 2D but it's what I was familiar with. For the actual physics simulations, I'm using matter.js.
The diagrams use a library I wrote while working on the post [1].
Everything runs inside a Vue app which serves as the glue between things like the sliders and the three.js scenes and diagrams.
Sounds intriguing! I’m probably your ideal audience, given a large coding background, low math background, and current writing 2D focused games.
If I may ask a question, in your mind what’s the benefit of more deeply understanding these implementations when physics frameworks handle a lot of the really mathy logic behind collisions and simulations.
Either way I’m putting this in my read queue. Thanks!
I'm curious whether you've considered using Box2D[1] as a reference, since it is a pretty complete implementation of rigid body physics? Or is it more that you want to re-implement things from scratch as part of this tutorial?
Oh! Look, a well-researched, deeply-explained, and interactive post.
Honestly, when I initially read the domain name and noticed the TLD is ".ski" I was thinking it is from the author who wrote about Mecanical Watch [1] and other cool stuff.. it turned out to be a totally different one but of similar quality. What's the secret sauce behind this ".ski" TLD :)
There is very simple reason for this: "ski" it the most common suffix of Polish surnames, of which the most popular is Kowalski. There are a lot of people out there either Polish or with Polish origins.
Author of https://ciechanow.ski – the site we all love here – is Polish programmer working for Apple.
On the same note there are some names that no matter the domain you know you're going to get a wizard, anyone named Thiago is a beast and I've chanced on several.
One side project I am working on right now is a 2d space shooter I am developing with my son. The idea is to have top down look, have each player control some kind of ship and fly in an enclosed area filled with space debris and shot opponents. An important aspect of this game is that the space debris can be moved around the arena and used creatively to capture opponents, prevent them from achieving their goals, etc.
As part of the project I was thinking we will skip game engine altogether because I wanted to teach my son a bit more about how to structure the application, etc. I thought in future we would use a ready game engine but at least once we would go through the exercise of implementing everything.
Everything was fine until we approached the problem of collision detection and handling. That's when things went downhill pretty quickly. Even with my theoretical math background I was soon consumed with just enormous amount of corner cases and finally decided to relent and use Box2d for this.
I am not a professional game developer, but I have over 20 years of development experience + math background and I still made the mistake of underestimating the problem. It really seems easy problem when you state it and just becomes exponentially more and more complex as you start digging into details.
Did your game require realistic physics collisions? If not, this might be unnecessary complexity. Almost no 2D shoot'em up game before 2000, and very few afterwards, go this route. Here's the common method to make a shmup with very simple rectangle comparisons:
But if your space debris objects are supposed to collide and agglomerate realistically, and if the player ships are supposed to have difficulty pushing a cluster of heavy objects out of the way, then using a physics library is sensible.
The point is you fly in a small spaceship in an arena, you hide behind obstacles, you shoot with a small variety of weapons and you use your weapons to either shoot the enemy directly or rearrange the map to make life difficult for your them. The collisions need to be realistic because you need to be able to predict what is going to happen when you hit things a certain way.
It is just a concept we are playing with.
Another part of that concept is that this game is meant for small kids that can't read. There is not a single letter or digit in the entire game. No menu. You just start the controller and get immediately pulled into the game.
And another feature is we wanted the game fun because the control feel fun and immediate. So we are experimenting a lot with what it means for the controls to be enjoyable.
Did you come across Verlet Integration [1]? It is quite convincing and usable for a lot of use cases, and is actually quite simple. I surprised myself when I was able to create a basic physics systems in a couple of hours with this great tutorial [2].
I don't know anything about licences, but no problem, help yourself, take the code, do whatever you want with it, sell it if you want, I would be pleased. Let me know what you built with it, it would be nice for me to see my code used in a game :)
Collisions are violations of the pairwise non-intersection constraint between bodies. Collision forces are Lagrange multipliers of these constraints. Collision normals are the (normalized) partial derivatives of the constraint function wrt one of the body's configurations.
That sounds like the kind of thing which works if you're doing physics at 1kHz+ with an integration algorithm with good numeric stability which honours conservation of energy, but in games, we're often running physics at down to 30Hz using some ad-hoc Euler-Chromer, which requires very different approaches
It's still the same principle even in games. If you are trying to explain where forces come from and how resolution works, you need to ground it in something. Otherwise you are just adding extra assumptions onto assumptions.
In proper physics simulations, everything's about forces, most things are springs, and you never teleport stuff. In games, modelling the ground as a spring really doesn't work and teleporting entities when they collide with parts of the world often makes a lot of sense. It's just not the same.
EDIT: If I'm incorrect, please explain how. I've written some game physics systems and seen some proper physics sim systems and these comments reflect my understanding of the situation, and if I've said something wrong, please correct me instead of just downvoting.
The principle of least constraint is the basis for rigid body mechanics based contact forces. This has been known since the days of Gauss and Hamilton, and is fundamentally how restitution and collision forces are derived in Lagrangian mechanics. There's a long literature on this going back more than a hundred years.
It's true that some commercial solvers like Ansys use spring/penalty methods, but this is due to the spring forces being easier to couple to other solvers. It's harder in the Ansys force/velocity formulation to combine things like elasticity and fluids to their rigid body solver. To deal with the instability of systems of many stiff springs they have to take many small timesteps to avoid convergence issues.
More recently techniques like XPBD have been gaining popularity, particularly in film, which use purely positional constraints and variational methods to combine many different types of physics simulations. There's a really great and approachable series of videos by Matthias Muller on youtube which goes through how to implement all this in JS https://matthias-research.github.io/pages/
Finally it's funny you should mention games, since many older games used spring methods for physics. It was only when constraint based solvers became popular after Havok/Half-Life 2 did we start to see games with real rigid body dynamics and stable stacking of boxes. Older physics games like Trespasser ( https://en.wikipedia.org/wiki/Trespasser_(video_game) ) had many bugs due to the use of hacky spring physics. For a good explanation of how games do it today look at Erin Catto's work on Box2D https://box2d.org/publications/
Most games since Half Life 2 use constraint forces like this to solve collisions. Springs/penalty forces are still used sometimes in commercial physics solvers since they're easier to couple with other simulations, but they require many small timesteps to ensure convergence.
This is really cool! Love the explanations, the interactions, and especially the friendly tone of voice of the article. Looking forward to the ones following.
Making a 2d rigid-body physics engine is a really fun project. I made one myself in javascript before learning about linear algebra. And I dug deep into the maths to get it working. Despite months of work, I barely scratched the surface beyond the widely known basics.
Making a stable engine where objects don't compress into one another or jitter is a rabbit hole without a bottom that even the most math-heavy articles I could find rarely touched.
To learn JS I started with canvas, and without any game dev experience, made a couple of cute little browser games. One is a galaga clone that, for the most part, works OK. The hard part is the projectile collisions. Instead of taking the position of your bullet now, and where it would be in the next time step, and seeing if it would intersect with the enemy hitboxes taken in the same way, I _only_ checked on the current timestep, which means your bullets can magically go around the enemies! Silly stuff. Maybe some day I'll go back and fix it.
Wow, perfect timing. Some days ago I started to get interested in this, particularly in circle collisions. Keep it up, looking forward for the rest of the posts.
This is great. This reminds me of Chris Hecker's Rigid Body Dynamics series from GDMag/Gamasutra that I read (checks watch) almost 30 years ago! This is the classic/canonical set of articles.
Guess this is a Shameless plug but I wrote an interesting program over 12 years ago using even then very old three Js which is not quite the metal but much less abstract than today's tools. I'm always amazed when I check it and it's still running
My own attempts at 3d games failed to get good sweeping polygon polygon collision to work. IIRC I found one good demo video of a programmer demonstrating it on youtube but no code or articles or anything.
I do wish this kept going and went "and as a last step, we need to add in torque" because pretty much everyone stops at regular force vectors, but torque transfer is so ridiculously important.
I know this post is focused on collision resolution, but I was still a little disappointed to see collision detection handwaved away. In my experience that's where the real headscratchers are.
Completely disagree - convex collision detection is a neatly solved problem. SAT [0] is implementable in a couple of hours, and GJK [1] is a really neat algorithm that takes a ltitle more than SAT to implement. For optimising collision detection there's some spatial partitioning [2] algorithms that work really well.
The gnarly stuff comes in when you're trying to figure out contact points and impulse responses and apply them without blowing everything up. I've written some of this stuff for a few games now, and the resources below have stood the test of time IMO
Well yeah, people often assume that because they know how to tell if two shapes are currently intersecting, they've solved collision detection...and then their games turn out to have a bunch of detection-related bugs and glitches. Most notoriously, lots of homebrewed physics libraries suffer badly from tunnelling (i.e. when one or both bodies are moving so fast that they completely pass through another before the next time collision is naively checked, and so the checker fails to detect a collision at all).
And on top of that there is more to collision detection than just code IMO - setting up hitboxes that both feel right for the player and aren't visually glitchy is not as straightforward as it might seem, especially with 3D.
(Also, figuring out contact points is part of competently implemented collision detection.)
I mean there's a lot of material out there already about collision detection though. This feels a bit like seeing a blog post about register allocation and being sad that it doesn't talk about parsing?
The way this article is written with small simulations throughout the page and word highlights was reminding me of https://ciechanow.ski/ and then I noticed the domain name. It is very similar too https://www.sassnow.ski/
Couldn't be happier seeing more people going to same depths (or heights) to explain things to people. Will be looking forward to more articles like this as this just the very first one.
With a sample size of 2, I declare that the .ski domain is dedicated to very high-quality interactive education websites. I am now conditioned to click on any .ski domain that I come across on HN.
Bartosz was definitely my biggest inspiration when writing this post. I was debating if I should actually use that domain because it might come across as a little too derivative but I couldn't resist!
To give some context, this is only part one in a series of blog posts I plan on writing about rigid body physics.
The post is aimed at people like myself, who aren't game devs and don't necessarily have a strong math background. Which is why I spend so much time explaining concepts that would appear almost trivial to someone who has experience in this area.
Happy to answer any questions you might have.