twisting_edge said:
Actually, that second one is a pretty interesting problem. You have all the information necessary to solve it, although the solution might be a little more involved than you think.
I can't sleep, and my mind has chosen this one to pick on.
It
is a somewhat interesting problem, but not very in the specific case. A more general solution is probably insoluable. But some generalities might be fun to review. Either that, or my brain is seriously misfiring at this late hour (almost certainly the latter).
In the case of identical raptor speed, sufficently low relative victim speed, the victim cannot cross the border of the triangle. His best move is to remain still. That much is obvious. We'll also assume instantly accelerating raptors smart enough to aim where the victim will be (as opposed to where he is) at this time, although it is not relevant in this case.
With one wounded raptor, his best move is to run (possibly not at full speed) toward the wounded raptor. By symmetry, this is also a straight line.
With equal raptor speeds and less disparate predator/prey speeds, the victim can cross the line delimiting the initial raptor triangle. His best move is full speed towards a boundary, and straight on from there. Once again, line remains straight by symmetry.
With one wounded raptor and less disparate speeds, I am fairly sure his best path becomes elliptical some critical distance after crossing the boundary. It may be a circular segment. But I think the condition of fully anticipatory raptors is starting to break down in that analysis.
I am fairly sure making the raptors run towards his current (as opposed to future) position yields an elliptical (or other conic section) post-critical path.
Once you allow for finite raptor acceleration, it's starting to look a lot like the three body problem, only not quite so bad. I suspect there's still a discontinuity in the second derivative of the victim path at a critical point outside the triangle.
Anyone care to run with this line of speculation from here (and away from the raptors)? Triple points will be taken down in evidence against you should you actually put pencil to paper and start to work out the math.
P.S.: The raptors may have to slow at times depending on the curvature of their respective paths at that point.
