alt Hacker NewsHm. Back when I was working on game physics engines this might have been useful.
In impulse/constraint mechanics, when two objects collide, their momentum changes in zero time. An impulse is an infinite force applied over zero time with finite energy transfer. You have to integrate over that to get the new velocity. This is done as a special case. It is messy for multi-body collisions, and is hard to make work with a friction model. This is why large objects in video games bounce like small ones, changing direction in zero time.
I wonder if nonstandard analysis might help.
Replies
Making it work in finite but short time should fix that. A large object generally can deform a larger distance. This makes all collisions inelastic, with large ones being different than small ones.
If you can get realistic billiards breaks, you're on the right track.
Nonstandard analysis is the mathematical description of your special case. Same thing.
The following is just my opinion:
Integration can be done with its own special arithmetic: Interval arithmetic. I base this suggestion on the fact that this is apparently the only way of automatically getting error bounds on integrals. It's cool that it works.
NSA does not work with a computable field so it's not directly useful. But at the end of the article, there's a link to some code that uses the Levi-Civita field, which is a "nice" approximation to NSA because it's computable and still real-closed. You might be able to do an "auto-limit" using it, in a kind of generalisation of automatic differentiation. This might for instance turn one numerical algorithm, like Householder QR, into another one, like Gaussian elimination, by taking an appropriate limit.
I don't know if these two things interact well in practice: Levi-Civita for algebraic limits and interval arithmetic for integrals. They might! This might suggest rather provocatively that integration is only clumsily interpreted as a limit of some function. Finally tbh, I'm not sure if this is the best solution to the friction/collision detection problem you're describing.