How far-fetched is 4D collision detection?

[kyle]

I'm curious as to whether or not anyone has ever considered this seriously. I'm sure someone has, but I can't really find anything on it; what if collision detection was processed in 4 dimensions?

For our purposes, we can treat time as an orthogonal 4th dimension, right? And at least GJK generalizes to four dimensions pretty smoothly, right?

So what are the real hurdles to implementing collision detection in four dimensions? Is there something I'm overlooking that'd make it slower than the current method of continuous collision detection, or is this already being done? Just wondering. If no one has any objection to it, I'll give it a try and see how it works.

But if anyone has a reason why this shouldn't work, please stop me before I waste my time.

[kyle]

Posting this as a reply for others:

It likely won't work because the 4-dimensional object will be concave if it rotates from time T1 to T2. Trying to figure out new convex hulls from this pretty much negates any benefit from before, so this is probably a futile path.

Interesting discussion points though?

[d3x0r]

A hypersphere or hypercube or other hyper primitive I'd think would remain convex in all rotations.

Though with time as a representation, I don't know that it would be that useful... a cube is here then it's there... at some later point it returns to 'here' does that mean it collided? does it matter? which direction would it have to go to resolve the collision?

Kinda of a reverse XY problem? you have a thing to think about, but don't really know what to do with it?

while all the math isn't that complex to extend to a 5x5 matrix for rotations, 5 points for a 'quinternion' and 4 values in vector ....

but what IS a rotation in 'time'?

[Basroil]

but what IS a rotation in 'time'?

There's an interesting "answer" to that http://casa.colorado.edu/~ajsh/sr/wheel.html

*The following is a cursory reading of that and related topics, it is most certainly incomplete and may be entirely wrong*
Apparently you can consider a rotation in (imaginary) time as a speed component, which in itself could be useful for something, though that math is going to get out of hand really quickly unless you take everything at face value and don't try to understand it. Optimizations are also going to get messy, since you won't know what to optimize (aside from standard matrix multiplication optimizations)

[d3x0r]

... from that link...

The − sign instead of a + sign in front of the t2 in the spacetime separation formula s2=−t2+x2+y2+z2 means that time t can often be treated mathematically as if it were an imaginary spatial dimension. That is, t=iw, where i is the square root of −1 and w is a “fourth spatial coordinate” (so −t2=w2).

as if squaring time makes it space. That's where the math all went wrong... yes i^2 is -1... which would then be a spacial coordinate... but it's not it's still just time. *shrug*

[kyle]

Interesting points. Indeed, a hypercube or hypersphere both remain convex, and I should've mentioned that I was thinking of convex hulls.

I'm still learning about physics, so I'll keep these things in mind for the time when I have a sufficient foundation!