Timeline for Curves of constant curvature on an ellipsoid
Current License: CC BY-SA 3.0
19 events
| when toggle format | what | by | license | comment | |
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| Sep 13, 2017 at 16:27 | comment | added | Narasimham | These trajectories depend strongly on initial conditions. | |
| Sep 13, 2017 at 16:11 | comment | added | Joseph O'Rourke | @Narasimham: 2:1:1. See the explicit equation of the ellipsoid in my post. | |
| Sep 13, 2017 at 16:09 | comment | added | Narasimham | What were axes proportions you have used? I am almost sure analytical $ r- \theta $ relation involves elliptic integrals whose periodicity determines closed geodesics formation. | |
| Jul 24, 2017 at 17:36 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
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| Jul 24, 2017 at 13:10 | answer | added | Robert Bryant | timeline score: 10 | |
| Jul 24, 2017 at 11:55 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
Image links broken; now fixed.
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jul 24, 2012 at 0:37 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
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| Jul 23, 2012 at 11:57 | comment | added | Joseph O'Rourke | @Ben: You were right to be suspicious! My method of computing the angular turn at each point of the curve was incorrect. I agree that closure should not be so prevalent. | |
| Jul 23, 2012 at 9:58 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
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| Jul 23, 2012 at 5:07 | comment | added | user21349 | It's strongly counterintuitive to me that randomly chosen initial conditions would lead with nonzero probability to a simple closed curve. For the curves you showed, did you choose initial conditions that had special symmetry? Did you verify with high precision, or only visually, that they returned to the same point with the same tangent vector? | |
| Jul 22, 2012 at 20:42 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
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| Jul 22, 2012 at 20:19 | comment | added | user21349 | In the degenerate case of a cylinder, you can clearly get non-simple closed curves curves. Draw a circle on a piece of paper and roll the paper up into a tube so that it self-intersects. For a prolate ellipsoid whose long axis is long compared to the scale set by the curve's curvature, you should get the same behavior. | |
| Jul 22, 2012 at 18:35 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
(Not sure they are "inexact," but they are "naive.")
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| Jul 22, 2012 at 18:02 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
Added ellipsoid experiments.
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| Jul 21, 2012 at 0:13 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
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| Jul 20, 2012 at 17:59 | comment | added | Joseph O'Rourke | @Will: A geometric picture! | |
| Jul 20, 2012 at 14:58 | comment | added | Will Sawin | do you want a geometric picture or a mathematical formula? | |
| Jul 20, 2012 at 12:06 | history | asked | Joseph O'Rourke | CC BY-SA 3.0 |