GJK: Collision detection algorithm in 2D/3D

I should be writing a thesis about AI but got inside the collision rabbit hole so I have this fresh.

From the description of the algorithm you are doing I think you are thinking about Lin-Canny or V-Clip, which certainly may have that kind of numerical error problems.

GJK has also numerical problems but they are different. In principle it shouldn't be affected by coplanarity of several faces since you just need the vertex with highest support for a given direction. It could be a problem if you find the support point by hill climbing from vertex to vertex. GJK however does have numerical problems but they are of a different kind related to the degeneracy of the simplices it computes.

But you are so right about the subtetly of the problem: there is a very fine thread between infinite looping and incorrect answers. I have been bitten by this trying to implement geometric algos. There should be a special hell for people that output coplanar faces.

I know one of Bullet Physics/MuJoCo has the GJK, not remember which one. If anyone is curious I know of two Julia implementations:

https://github.com/JuliaRobotics/EnhancedGJK.jl

and my favorite: https://github.com/arlk/ConvexBodyProximityQueries.jl

This latter one is great as you are just required to implement the support function and are ready to go. Julia performance is great if you are concerned about using a dynamic language (i.e: ~2us for collision between two convex bodies of 1000 faces each)

Finally, about the convex hull computation it looks like some kind of solved problem, I mean, O(n log(n)) for 3D. Wrong!!!! QHull in this regard is fantastic as it has several heuristics to solve problems caused by finite precision, not to mention that I think worse case is O(n^2) as it doesn't implement the asymptotically optimal algo (not sure...). If you scale to more dimensions, which could happen even in if 3D because you transformed your problem to a convex hull problem you will be hit with O(n^2), bad news. There are several other libraries (CCD, LRSLib and more) that allow you to use arbitrary precision but you will get something like a 100x penalization for the luxury.

Nice, I didn't know about Jolt.

A good book also is "Real Time Collision Detection" (Christer Ericson).

As the other comments note, you can create a sweep form to detect penetration. But this belies the more thorny issues of achieving a "high quality" solution: if two things impact a third within the exact same time step, you have to come up with a globally satisfying solution that coordinates all three, and sweeping doesn't do that by itself, it just says "well at some time it might hit one or both of them" - which if you run with directly, leads to weird behaviors related to the information loss where objects squeeze past each other or gain excess energy or trigger events they shouldn't. The solution to that problem is generally handled through the solver by various iterative methods. In a 2D platforming game using AABBs this is as simple as "step the physics on Y axis, then step them on X axis" - which completely eliminates the possibility of hitting two things at once, albeit it makes corners "sticky" on one axis. Once you have things rotating and colliding at any angle it gets a lot harder and the focus then is on how to iterate approximately in a way that produces the least error. (If you can afford it, backtracking the solver to optimize the solution quality to some metric of fewest/smallest impacts is one way to create satisfying interactions).

This is actually a fairly common problem in game development! There are two largely used solutions: 1. Don't have fast moving objects 2. Use "continuous" collision detection.

Option 2 is more interesting. Continuous collision detection is, in essence, "looking ahead" at where an object is going and seeing if there are any potential collisions (typically using a bounding box or simplified geometry to save on CPU power). If there are no potential collisions, all is well, and if there is a potential collision we can then calculate its outcome.

One option is to not worry about it so speedrunners can have a bit of extra fun breaking the game.

It's a feature, not a bug!

The usual approach is some form of sweep to get a time of impact. Once you've got a time of impact, you can either generate contacts, or avoid integrating the involved bodies beyond the time of impact, or do something fancier like adaptively stepping the simulation to ensure no lost time.

If the details don't matter much, it's common to use a simple ray cast from the center at t0 to the center at t1. Works reasonably well for fast moving objects that are at least kinda-sorta rotationally invariant. For two dynamic bodies flying at each other, you can test this "movement ray" of body A against the geometry of body B, and the movement ray of body B against the geometry of body A.

One step up would be to use sphere sweeps. Sphere sweeps tend to be pretty fast; they're often only slightly more complicated than a ray test. Pick a sphere radius such that it mostly fills up the shape and then do the same thing as in the previous ray case.

If you need more detail, you can use a linear sweep. A linear sweep ignores angular velocity but uses the full shape for testing. Notably, you can use a variant of GJK (or MPR, for that matter) for this: http://dtecta.com/papers/jgt04raycast.pdf

If you want to include angular motion, things get trickier. One pretty brute forceish approach is to use conservative advancement based on distance queries. Based on the velocity and shape properties, you can estimate the maximum approaching velocity between two bodies, query the distance between the bodies (using algorithms like GJK or whatever else), and then step forward in time by distance / maximumApproachingVelocity. With appropriately conservative velocity estimates, this guarantees the body will never miss a collision, but it can also cause very high iteration counts in corner cases.

You can move a lot faster if you allow the search to look forward a bit beyond potential impact times, turning it into more of a root finding operation. Something like this: https://box2d.org/files/ErinCatto_ContinuousCollision_GDC201...

I use a combination of speculative contacts and then linear+angular sweeps where needed to avoid ghost collisions. Speculative contacts can handle many forms of high velocity use cases without sweeps- contact generation just has to be able to output reasonable negative depth (separated) contacts. The solver handles the rest. The sweeps use a sorta-kinda rootfinder like the Erin Catto presentation above, backed up by vectorized sampling of distance. A bit more here, though it's mainly written for users of the library: https://github.com/bepu/bepuphysics2/blob/master/Documentati...

In games the fastest moving object is bullets, so model them as lines, not points.

Yep! I saw it and stumbled upon this pretty blog post.

I'm implementing this right now.

Trying to figure out a way to render the support functions to assist with debugging my implementation. At the moment I just sample the function in various directions around a unit sphere and try build a mesh from that. Too patchy.

Another nice thing about this one is that in the case where the objects intersect, you can just keep iterating mostly the same algorithm to get the intersection points, rather than needing to use Expanding Polytope as with GJK.

This is the explanation that made it click for me, although in my experience it seems like he's wrong about being able to rule out the need for doing certain checks when you already "know" the origin is not in that section due to how you got to where you are. Not sure if my implementation was just shitty, but you'll get weird results when objects are very close if you don't exhaustively check every sector - probably due to numerical imprecision, if I had to guess.

You can use GJK for 2D also. You don't need to compute the bounding boxes as GJK will automatically "convexify" the body although if you "precompute" the bounding boxes it will be better. You just need to compute the support function in an arbitrary direction.

If you want to use bounding boxes then you can do it faster using the separating axis theorem.

I don't understand what you mean