I suggest you take your objections up with the authors of, say,

A Four-Dimensional Space-Time Automatic Obstacle Avoidance Trajectory Planning Method for Multi-UAV Cooperative Formation Flight

or any number of other similar papers.

Many prefer to think of an objects path as a trajectory in space-time (four dmensions) and for two ojects to "collide" their paths must coincide within that 4-D space within an Epsilon for some value of WTF.

Other common examples of working in N dimensional spaces for values of N bigger than someones mother include: https://en.wikipedia.org/wiki/Phase_space

I never considered time as a 4th dimension in that sense. TIL.

Fun fact, now I can claim to plan in the 4th dimension by using lead times in supply chain planning!

But as ... you yourself ... note above, the planes aren't moving in four dimensions. If time is one of the dimensions, all the planes are doing is existing.

A plane (P1)'s past positions and projected future positions form a continuous path in R^4 (four dimensional euclidean space).

Another plane (P2)'s travel through various {X,Y,Z} positions at various times forms another path in R^4.

If those paths come close to each other then P1 and P2 are close in both space and time - ie. they are very close to a collision.

Collision detection and avoidance is problem laid out and (hopefully) solved in an R^4 euclidean space.

(at the very least - throw in some more independant variables that parameterise motion and you've got a higher order puzzle

Eg: Collision avoidance for two robot arms with 6 or 7 degrees of freedom each is a maze solving puzze in 12 or 14 dimensions).

Where do you think you're contradicting me?

The path of the plane is a static curve in 4-dimensional space, yes.

But the plane is not located at any point in the 4-dimensional space, and the position in 4-dimensional space that it doesn't have is not changing over time. Both of those things are required before you can describe the plane as "moving" within the space.

There is no secret backup time that will allow you to track the plane's hypothetical motion along an explicit time dimension. That's not a thing.

> But as ...

I confess. I literally had no idea what you were intending to convey with those two sentences so I restated alternatively what I intended to convey in the hope it might make clear my position (if that was an issue for you) or that I might learn more from your response.

> Where do you think you're contradicting me?

That's not a thought that I thunk. Therefore I have no response.

> But the plane is not located at any point in the 4-dimensional space

Every point along the 4D path trajectory of the plane in {X,Y,Z,T} is a literal {X,Y,Z} location of that plane at time T.

> Both of those things are required before you can describe the plane as "moving" within the space.

I certainly did not describe the plane as "moving" within R^4.

> There is no secret backup time that will allow you to track the plane's hypothetical motion along an explicit time dimension.

I utterly fail to understand what you intend to convey here.

Although I note that the actual (not hypothetical) velocity of the plane projected onto the time axis is very likely to be on the order of approximately one second per second.