3 ms·

For those curious about the mathematical objects involved: http://mathworld.wolfram.com/PursuitCurve.html http://mathworld.wolfram.com/PursuitCurve.html https:

For those curious about the mathematical objects involved: http://mathworld.wolfram.com/PursuitCurve.html http://mathworld.wolfram.com/PursuitCurve.html https://en.wikipedia.org/wiki/Pursuit_curve https://en.wikipedia.org/wiki/Pursuit_curve If you model this, you will probably get very different answers based on how well you discrete-ize (is there a better word for that idea?) the time and movement. Also, it's not a turn based game, i.e. the man doesn't move and then the lion moves. They move together at the same time. I think if you could answer what happens if the man has his back to wall and the lion is as close as possible without catching him, then you would solve the problem. Because the lion has to predict where the person is going to move next, it feels like you won't ever know for sure what will happen. As time tends towards infinity, the lion will probably eat the guy due to sheer chance of both of them going the same direction at the right time.

aparadja 15y ago
Related image from your first link http://mathworld.wolfram.com/images/gifs/purscir4.gif http://mathworld.wolfram.com/images/gifs/purscir4.gif
- bradleyland 15y ago
  My problem with this representation is that it assumes the pursuer is unable to "lead" its target. The pursuit curve, as modeled in mathematical representations constrains the pursuer to traveling only on a path that points directly at the pursued. Lions are highly adapted hunters that frequently take down prey that are much faster than them. It would seem that they would have to develop strategies for optimizing the distance-to-intercept in order to be successful as a species. Thus, I believe the man is eaten rather quickly, even if the two are equal in speed.
  - gcp 15y ago
    My problem with this representation is that it assumes the pursuer is unable to "lead" its target. How can the pursuer lead the target when the target moves randomly? At best, he can adjust his lead with an infinitesimal delay, but its not clear this allows him to catch the target in finite time.
    - bradleyland 15y ago
      And yet Lions catch faster prey all the time. Just because something occurs randomly doesn't mean it always results in a net loss for the pursuer. All the time spent in pursuit on an optimized (leading) path results in closed distance. Within the constraints of the cage the pursued can only choose to zig or to zag, so if the lion uses a leading path, and is correct in anticipating directional changes only half the time, he's still going to have lunch sooner rather than later. There are visual "tells" that allow the lion to anticipate directional shifts on the part of their prey, so I'd argue that his percentages will be significantly better. I guess I'm struggling with understanding the value of the exercise. If we're talking inanimate bodies, I get it. This kind of physics is valuable. However, choosing arbitrary constraints in order to make interesting maths out of natural scenarios is low hanging fruit. I just don't see it as all that interesting.
karamazov 15y ago
Letting time go to infinity won't affect the answer - if the tamer has a strategy that allows him to avoid getting eaten, he's going to implement it all times, rather than risk making random movements.
- zackzackzack 15y ago
  I was thinking in terms of the man making random movements at some point. I don't really believe there is a strategy that allows him to avoid getting eaten so the best thing he could do is random movements.
beza1e1 15y ago
The pursuit curve assumes that the lion always moves directly towards the tamer. The better strategy for the lion is your back-to-the-wall position. The lion should be able to get infinitesimally close to the tamer. Pragmatically and finally, the delta will be smaller the reach of the lion's pawns. Lion wins.
- amirmc 15y ago
  "...back-to-the-wall position." Can you expand on this? My thought was that if the lion 'zig-zagged' somewhat, it would effectively force the tamer to constantly change direction[1] and ultimately the lion would catch up. Is that what you meant by back-to-the-wall? [1] Assuming that the tamer's only strategy is to maximise distance from the lion.
  - beza1e1 15y ago
    Yes, I also have the zig-zag in mind. However, as a point, the lion will never catch him. The lion has to move further to the side than the tamer, so the tamer switches directions after each "step". This works, because within a circle the tamer with his back to the wall has to move a little towards the lion while moving sideways. The lion moves a little bit more to the side a little bit less towards the tamer. Over time the lion comes closer and the steps become smaller. When it gets really close however, the round wall looks increasingly like a flat wall. Lion and tamer end up just moving sideways without ever touching? We need some more math for this.
gcp 15y ago
As time tends towards infinity, the lion will probably eat the guy due to sheer chance of both of them going the same direction at the right time. This is all but obvious to me. I pick a random number from 0 to 2*Pi. If you guess my number correctly, you get to eat me. You can guess an infinite amount of times. Who wins this game? Does the answer depend on the assumption of fundamental axioms?
- deleted 15y ago
  [deleted]
hythloday 15y ago
is there a better word for that idea? Quantize. :)
- zackzackzack 15y ago
  Thank you! I knew it had to be a word that I knew and had shamefully forgotten.

aparadja 15y ago

Related image from your first link http://mathworld.wolfram.com/images/gifs/purscir4.gif http://mathworld.wolfram.com/images/gifs/purscir4.gif

bradleyland 15y ago
My problem with this representation is that it assumes the pursuer is unable to "lead" its target. The pursuit curve, as modeled in mathematical representations constrains the pursuer to traveling only on a path that points directly at the pursued. Lions are highly adapted hunters that frequently take down prey that are much faster than them. It would seem that they would have to develop strategies for optimizing the distance-to-intercept in order to be successful as a species. Thus, I believe the man is eaten rather quickly, even if the two are equal in speed.
- gcp 15y ago
  My problem with this representation is that it assumes the pursuer is unable to "lead" its target. How can the pursuer lead the target when the target moves randomly? At best, he can adjust his lead with an infinitesimal delay, but its not clear this allows him to catch the target in finite time.
  - bradleyland 15y ago
    And yet Lions catch faster prey all the time. Just because something occurs randomly doesn't mean it always results in a net loss for the pursuer. All the time spent in pursuit on an optimized (leading) path results in closed distance. Within the constraints of the cage the pursued can only choose to zig or to zag, so if the lion uses a leading path, and is correct in anticipating directional changes only half the time, he's still going to have lunch sooner rather than later. There are visual "tells" that allow the lion to anticipate directional shifts on the part of their prey, so I'd argue that his percentages will be significantly better. I guess I'm struggling with understanding the value of the exercise. If we're talking inanimate bodies, I get it. This kind of physics is valuable. However, choosing arbitrary constraints in order to make interesting maths out of natural scenarios is low hanging fruit. I just don't see it as all that interesting.

karamazov 15y ago

Letting time go to infinity won't affect the answer - if the tamer has a strategy that allows him to avoid getting eaten, he's going to implement it all times, rather than risk making random movements.

zackzackzack 15y ago
I was thinking in terms of the man making random movements at some point. I don't really believe there is a strategy that allows him to avoid getting eaten so the best thing he could do is random movements.

beza1e1 15y ago

The pursuit curve assumes that the lion always moves directly towards the tamer. The better strategy for the lion is your back-to-the-wall position. The lion should be able to get infinitesimally close to the tamer. Pragmatically and finally, the delta will be smaller the reach of the lion's pawns. Lion wins.

amirmc 15y ago
"...back-to-the-wall position." Can you expand on this? My thought was that if the lion 'zig-zagged' somewhat, it would effectively force the tamer to constantly change direction[1] and ultimately the lion would catch up. Is that what you meant by back-to-the-wall? [1] Assuming that the tamer's only strategy is to maximise distance from the lion.
- beza1e1 15y ago
  Yes, I also have the zig-zag in mind. However, as a point, the lion will never catch him. The lion has to move further to the side than the tamer, so the tamer switches directions after each "step". This works, because within a circle the tamer with his back to the wall has to move a little towards the lion while moving sideways. The lion moves a little bit more to the side a little bit less towards the tamer. Over time the lion comes closer and the steps become smaller. When it gets really close however, the round wall looks increasingly like a flat wall. Lion and tamer end up just moving sideways without ever touching? We need some more math for this.

gcp 15y ago

As time tends towards infinity, the lion will probably eat the guy due to sheer chance of both of them going the same direction at the right time. This is all but obvious to me. I pick a random number from 0 to 2*Pi. If you guess my number correctly, you get to eat me. You can guess an infinite amount of times. Who wins this game? Does the answer depend on the assumption of fundamental axioms?

deleted 15y ago
[deleted]

hythloday 15y ago

is there a better word for that idea? Quantize. :)

zackzackzack 15y ago
Thank you! I knew it had to be a word that I knew and had shamefully forgotten.