Timeline for Plane projection of Geodesics (Inverse view)
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 22, 2022 at 14:31 | answer | added | Anton Petrunin | timeline score: 1 | |
| Apr 13, 2019 at 7:03 | history | edited | Shahrooz | CC BY-SA 4.0 |
The question became more exact
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| Apr 9, 2019 at 23:56 | comment | added | Anton Petrunin | Well, then you should change the question (others do not understand it still). | |
| Apr 8, 2019 at 19:47 | comment | added | Ben McKay | I believe that it is known (probably from Grayson's work on shrinking of curves) that, for any embedded closed smooth curve in the plane, there is a diffeomorphism of the plane taking that curve to a circle. From there you can be quite explicit, mapping the plane to, for example, a punctured sphere, so that the circle is mapped to a great circle, by projection from the north pole of the sphere (Ptolemaic projection). | |
| Apr 8, 2019 at 19:34 | comment | added | Shahrooz | @Ben McKay: I can imagine what you say, but how can we prove that such bending is possible and the resulting projected curve on the light bulbe shape is a geodesic? | |
| Apr 8, 2019 at 19:29 | comment | added | Shahrooz | @Anton Petrunin: It seems that by your suggestion the problem will be more exact. I agree with the orthogonal projection and closed surface. Is there any positive answer in this case? | |
| Apr 8, 2019 at 18:25 | review | Close votes | |||
| Apr 9, 2019 at 9:30 | |||||
| Apr 8, 2019 at 18:06 | comment | added | Anton Petrunin | The formulation is very unclear. Did you want say that $\gamma$ is orthogonal projection to the plane and $M$ is a closed surface? | |
| Apr 8, 2019 at 16:00 | comment | added | Ben McKay | You can take your simple curve (if it is embedded) to a circle by a diffeomorphism of the plane, which is the identity outside a compact set, and then bend your plane into a light bulb shape. Is that what you are looking for? The details of such an argument would be long, I suppose. | |
| Apr 8, 2019 at 13:44 | comment | added | Shahrooz | @JosephO'Rourke Thanks for the link. It seems interesting and helpful! | |
| Apr 8, 2019 at 13:40 | comment | added | Joseph O'Rourke | Related: Is every closed curve in 3D a geodesic on a genus-0 surface?. | |
| Apr 8, 2019 at 13:32 | history | edited | Shahrooz | CC BY-SA 4.0 |
added 51 characters in body
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| Apr 8, 2019 at 13:25 | history | asked | Shahrooz | CC BY-SA 4.0 |