A fast 3D collision detection algorithm
16 comments
·July 9, 2025Animats
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.
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
d ≥ |x-y|^2
x ∈ A
y ∈ B
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
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.
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
reactordev
Without optimizations of the vertices buffer, correct, it’s a 1T loop. But we can work on faces and normals so that reduces it by a factor of 3. We can octree it further as well spatially but…
There’s a really clever trick Unreal does with their decimation algorithm to produce collision shapes if you need to. I believe it requires a bake step (pre-compute offline).
I’d be fine with a bake step for this.
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.
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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.
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
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MortyWaves
I’m getting sick of the number of links submitted to HN blasting me with cookie spam bullshit.
I discovered this collision detection algorithm during COVID and finally got around to writing about it.
github repo: https://github.com/cairnc/sat_blog