Timeline for Do curvature differences obstruct a.e orientation-preserving isometries?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Apr 14, 2017 at 20:56 | comment | added | Artur Jackson | Well nice find then! | |
| Apr 14, 2017 at 17:57 | comment | added | Anton Petrunin | @ArturJackson they are stolen :) | |
| Apr 14, 2017 at 5:32 | comment | added | Artur Jackson | How did you do those nice cartoons? | |
| Mar 18, 2017 at 20:50 | comment | added | Anton Petrunin | @AsafShachar "cut" means "cut", with scissors --- look at the picture; yes geodesic triangles will work for you; I am not sure what do you mean by "accurate", I would say "fractal". | |
| Mar 18, 2017 at 19:39 | comment | added | Asaf Shachar | I am still not sure what exactly do you mean by the cuts and the tree. Even more basically, by polygons on the sphere, do you mean union of geodesic triangles (or graphs that can be decomposed into a triangulation, that is perhaps with some of the edges missing?). If I understood correctly the spirit of your argument, you are trying to build a "very accurate" map of the sphere. (in the sense of geographic maps). | |
| Mar 18, 2017 at 14:43 | comment | added | Anton Petrunin | @AsafShachar Yes, I consider sphere with round metric. After cutting along the tree you can develop sphere into plane with very small distortion. However you may loose all this in the limit. It happens if you cut the sphere along the meridians from the north pole almost to the south pole. In order to keep the property, you need to cut along a fractal tree, so that most of the time one can travel between close points without going too far and passing cuts. | |
| Mar 18, 2017 at 8:20 | comment | added | Asaf Shachar | Thanks. However, I am afraid I do not understand too many parts in your idea; (1) Do you consider $\mathbb{S}^2$ with the round metric? (2) I am not sure what is the role of the maximal tree of cuts? (you said cuts of $1$-skeleton, so I guess you just mean "edges"? (3) By "development", do you mean to a "gluing diagram" (a set of polygons that is assigned the way in which they should be glued to each other along sides and vertices)? Is such a diagram always unique? (Perhaps this what the tree for...) | |
| Mar 17, 2017 at 23:48 | history | edited | Anton Petrunin | CC BY-SA 3.0 |
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| Mar 17, 2017 at 21:18 | history | edited | Anton Petrunin | CC BY-SA 3.0 |
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| Mar 17, 2017 at 21:05 | history | answered | Anton Petrunin | CC BY-SA 3.0 |