A fast 3D collision detection algorithm

[OlympicMarmoto]

I discovered this collision detection algorithm during COVID and finally got around to writing about it.

github repo: https://github.com/cairnc/sat_blog

[Animats]

Nice. It's definitely an optimization problem. But you have to look at numerical error.

I had to do a lot of work on GJK convex hull distance back in the late 1990s. It's a optimization problem with special cases.

Closest points are vertex vs vertex, vertex vs edge, vertex vs face, edge vs edge, edge vs face, and face vs face. The last three can have non-unique solutions. Finding the closest vertices is easy but not sufficient. When you use this in a physics engine, objects settle into contact, usually into the non-unique solution space. Consider a cube on a cube. Or a small cube sitting on a big cube. That will settle into face vs face, with no unique closest points.

A second problem is what to do about flat polygon surfaces. If you tesselate, a rectangular face becomes two coplanar triangles. This can make GJK loop. If you don't tesselate, no polygon in floating point is truly flat. This can make GJK loop. Polyhedra with a minimum break angle between faces, something most convex hullers can generate, are needed.

Running unit tests of random complex polyhedra will not often hit the hard cases. A physics engine will. The late Prof. Steven Cameron at Oxford figured out solutions to this in the 1990s.[1] I'd discovered that his approach would occasionally loop. A safe termination condition on this is tough. He eventually came up with one. I had a brute force approach that detected a loop.

There's been some recent work on approximate convex decomposition, where some overlap is allowed between the convex hulls whose union represents the original solid. True convex decomposition tends to generate annoying geometry around smaller concave features, like doors and windows. Approximate convex decomposition produces cleaner geometry.[2] But you have to start with clean watertight geometry (a "simplex") or this algorithm runs into trouble.

[1] https://www.cs.ox.ac.uk/stephen.cameron/distances/

[2] https://github.com/SarahWeiii/CoACD

[OlympicMarmoto]

> But you have to look at numerical error.

Yeah I agree, the error analysis could be many blogs in and of itself. I kinda got tired by the end of this blog. I would like to write a post about this in the future. For global solvers and iterative.

> Finding the closest vertices is easy but not sufficient.

As I'm sure you are aware, most GJK implementations find the closest features and then a one shot contact manifold can be generated by clipping the features against each other. When GJK finds a simplex of the CSO, each vertex of the simplex keeps track of the corresponding points from A and B.

> A second problem is what to do about flat polygon surfaces

Modern physics engines and the demo I uploaded do face clipping which handle this. For GJK you normally ensure the points in your hull are linearly independent.

[CamperBob2]

There's been some recent work on approximate convex decomposition, where some overlap is allowed between the convex hulls whose union represents the original solid.

I wonder if it would be smart to restate the problem in just those terms -- managing bounding-volume overlap rather than interpenetration at the geometric level.

If everything is surrounded by bounding spheres, then obviously collision detection in the majority of cases is trivial. When two bounding spheres do intersect, they will do so at a particular distance and at a unique angle. There would then be a single relevant quantity -- the acceptable overlap depth -- that would depend on the angle between the BV centers, the orientation of the two enclosed objects, and nothing else. Seems like something that would be amenable to offline precomputing... almost like various lighting hacks.

Ultimately I guess you have to deal with concavity, though, and then the problem gets a lot nastier.

[contingencies]

I am consistently amazed at your depth of knowledge. Can we have a meal? My shout. I'm crossing the US a lot for fundraising right now, probably nearby someplace. Email in profile.

[EGreg]

Nice to see this! I was writing about this topic 25 years ago: https://www.flipcode.com/archives/Theory_Practice-Issue_01_C...

And part two: https://www.flipcode.com/archives/Theory_Practice-Issue_02_C...

[msteffen]

I'm trying to work through the math here, and I don't understand why these two propositions are equivalent:

1) min_{x,y} |x-y|^2

   x ∈ A

   y ∈ B

2) = min_{x,y} d

   d ≥ |x-y|^2

   x ∈ A

   y ∈ B

What is 'd'? If d is much greater than |x-y|^2 at the actual (x, y) with minimal distance, and equal to |x-y|^2 at some other (x', y'), couldn't (2) yield a different, wrong solution? Is it implied that 'd' is a measure or something, such that it's somehow constrained or bounded to prevent this?

[OlympicMarmoto]

This is the epigraph form of the problem. You try to find the point with the lowest height in the epigraph.

https://en.wikipedia.org/wiki/Epigraph_(mathematics)

[Jaxan]

But why would d be much greater. The problem asks to minimise d, and so it cannot be greater than the smallest |x-y|^2.

[mathgradthrow]

I can't read substack on my phone, so I can't see the article, but the correct statement that is closest to what you have written is just that d is any real number satisfying this inequality. We define a subset U of AxBxR by

U={(a,b,x):x>|a-b|^2}

and then were looking for the infimum of (the image of) U under the third coordinate function

d(a,b,x)=x

[markisus]

I think you are missing that d, x, and y are variables that get optimized over. Any choice of d lower than the the solution to 1) is infeasible. Any d higher than the solution to 1) is suboptimal.

edit: I see now that the problem 2) is missing d in the subscript of optimization variables. I think this is a typo.

[leoqa]

Aside: I learned the Sep Axis Theorem in school and often use it for interviews when asked about interesting algorithms. It's simple enough that you can explain it to non-technical folks. "If I have a flashlight and two objects, I can tell you if they're intersected by shining the light on it". Then you can explain the dot product of the faces, early-exit behavior and MTV.

[ttw44]

This stuff is always so interesting to me because this is the part/step of solo game-development where people don't make too much content about it online, so you need to find books/scraps of content about rolling your own complex physics engine. It's all knowledge that's stuck in the industry/random github repos it seems.

[RobRivera]

Im reading 'Primer on 3D Math for Graphics Programmers"

I highly recommend it to you.

[reactordev]

This is novel indeed! What about non-spherical shapes? Do we assume a spherical bounds and just eat the cost? Either way, narrow phase gets extremely unwieldy when down to the triangle level. Easy for simple shapes but if you throw 1M vertices at it vs 1M vertices you’re going to have a bad time.

Any optimization to cut down on ray tests or clip is going to be a win.

[bob1029]

> Do we assume a spherical bounds and just eat the cost?

We pick the bounding volume that is most suitable to the use case. The cost of non-spherical bounding volumes is often not that severe when compared to purely spherical ones.

https://docs.bepuphysics.com/PerformanceTips.html#shape-opti...

Edit: I just noticed the doc references this issue:

https://github.com/bepu/bepuphysics2/issues/63

Seems related to the article.

[bruce343434]

Most likely this can be preceded by testing branches of some spatial hierarchy datastructure, 1 million squared is a lot to compute no matter the algorithm

[andrewmcwatters]

Usually you have a render model and a physical model which is a degenerate version of the viewed object, with some objects tailored for picking up, or allowing objects to pass through a curved handle, etc.

I would assume using this algorithm wouldn't necessarily change that creation pipeline.

[OlympicMarmoto]

Do you mean non-convex shapes? You can do a convex decomposition and then test all pairs. Usually games accelerate this with a BVH.

[socalgal2]

Related? This was posted a few days ago

https://news.ycombinator.com/item?id=44334403

[OlympicMarmoto]

No AVBD is a solver, they use standard collision detection routines.

[double051]

Hey that's Ascension from Halo 2. Cool test case!

[chickenzzzzu]

Real OGs know :) I used to love all of the super bounces and out of bounds tricks, though I think Ascension didn't really have many of the latter

[MortyWaves]

I’m getting sick of the number of links submitted to HN blasting me with cookie spam bullshit.

[MortyWaves]

-4 points suggests many HN users seem to enjoy being spammed by popups