Timeline for Herding sheep in a polygon
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Sep 29, 2017 at 20:12 | comment | added | Joseph O'Rourke | @NickGill: "In your example, do you assume after the first bit of barking at vertex $x$ that all the un-herded sheep are at point $b$?" No. I think of sheep balls rather than sheep points, and if they don't get through the gate, they would be stuck where they sit because there's another sheep in front of them. That's why I left that region above $xa$ still filled with sheep: There is nowhere for them to go. | |
| Sep 29, 2017 at 14:28 | comment | added | Gerhard Paseman | Another analytic approach would be to have the herding point x outside the polygon (use a really big dog), and consider when herding is successful as x approaches the polygon. It may have to do with when " parts of the convex hull are strictly contained with the polygon ", more accurately when all obtuse lines to a polygon boundary stay outside the polygon. Gerhard "Is Trying To Be Acute" Paseman, 2017.09.29. | |
| Sep 29, 2017 at 13:06 | comment | added | Adam P. Goucher | @CristiStoica Two opposite vertices of a square would also be non-herdible. | |
| Sep 29, 2017 at 12:32 | comment | added | Sebastian Goette | The two directions seem to occur only in isolated places. As there is always a little loss in real life, would you be happy to herd almost all sheep? | |
| Sep 29, 2017 at 12:10 | comment | added | Joseph O'Rourke | @NickGill: I must admit I didn't think of the two directions issue... (Cannot respond now.) | |
| Sep 29, 2017 at 12:04 | comment | added | Nick Gill | A point of clarification: In your example, do you assume after the first bit of barking at vertex $x$ that all the un-herded sheep are at point b? hence they can then be herded from point $x'$? This makes sense to me, I just want to check. Also... It's easy to imagine configurations where a sheep may have TWO directions in which it would move in response a dog's bark, so I guess in that case one would need to account for both? | |
| Sep 29, 2017 at 10:58 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
Clarifying model in response to Cristi Stoica's comments.
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| Sep 29, 2017 at 10:56 | comment | added | Joseph O'Rourke | @CristiStoica: Good suggestions re indeterminancy. I prefer the model where there is no sheep @$x$, and that's the only exception. So indeed they can get stuck in another corner. Will clarify. But I can see from your remarks there are other reasonable models. | |
| Sep 29, 2017 at 9:55 | comment | added | Cristi Stoica | @Gerhard Paseman: I believe this configuration with two gates is not herdible i.imgur.com/sUrwdQh.png | |
| Sep 29, 2017 at 9:25 | comment | added | Cristi Stoica | If a sheep is in x, then it seems there is an indeterminacy of the direction of motion. We can assume they are never in x, but they may arrive at a sharp corner, and the only way to herd them out of that corner is to move the dog there, so we run again into this indeterminacy. Maybe assume the choice is random? Another indeterminacy happens if the gate angle is >90 and x is adjacent to it: will the sheep choose the gate, or it will move along the next edge? Maybe assume it always chooses the gate. A simple example containing both these cases is a triangle with obtuse gate. | |
| Sep 28, 2017 at 18:47 | comment | added | Joseph O'Rourke | @GerhardPaseman: Any assumption that leads to clarity is welcome. I wanted to remove dynamics. So I assume the dog sits at a vertex and clears a region. Then moves to another and clears. And so on. | |
| Sep 28, 2017 at 18:32 | comment | added | Gerhard Paseman | Also, make sure your field is not fractal. I suspect members of (iterations toward ) the Koch snowflake are not k herdible for small k. Gerhard "Worse Than An Art Gallery" Paseman, 2017.09.28. | |
| Sep 28, 2017 at 18:28 | comment | added | Gerhard Paseman | Do you insist on one gate g? You might show every polygon is herdible through two gates that are (say) separated by at most one vertex. This extension might shed light on your classification. Gerhard "These Ways To The Egress" Paseman, 2017.09.28. | |
| Sep 28, 2017 at 13:34 | history | asked | Joseph O'Rourke | CC BY-SA 3.0 |
