Timeline for A necessary and sufficient condition for a space curve to lie on a ellipsoid
Current License: CC BY-SA 3.0
28 events
| when toggle format | what | by | license | comment | |
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| Mar 28, 2018 at 3:26 | answer | added | Mohammad Ghomi | timeline score: 8 | |
| Mar 27, 2018 at 9:21 | answer | added | Robert Bryant | timeline score: 19 | |
| Mar 26, 2018 at 14:23 | comment | added | Robert Bryant | It should be noted that the criterion that you write down is only necessary and sufficient for fully nondegenerate space curves, i.e., (connected) curves for which $\kappa$ and $\tau$ are nowhere vanishing. In fact, no local characterization can be both necessary and sufficient, since there exist (smooth, connected) space curves (even ones with $\kappa$ nowhere vanishing) such that every point of the curve has an open neighborhood within which the curve lies on a sphere, but the whole curve does not lie on any sphere. | |
| Mar 26, 2018 at 4:09 | history | reopened |
j.c. Willie Wong Stefan Kohl♦ Peter Michor Pace Nielsen |
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| Mar 25, 2018 at 3:01 | comment | added | Mohammad Ghomi | @user122298 The reason characterizing spherical curves is easy is that one can easily express the center of the osculating sphere in terms of curvature and torsion. Setting the derivative of that to zero, then yields the equations. I do not see an easy way to do it for ellipsoids. It may even be possible that there is no reasonable or pretty answer. At any rate, it is not a trivial problem. Just finding some necessary conditions would be quite interesting. | |
| Mar 25, 2018 at 1:50 | comment | added | Niven Zhao | @PeterMichor since affine transformation doesn't preserve the differential properties of curves, this approach may not be helpful to find the relations between curvature and torsion. | |
| Mar 25, 2018 at 1:26 | comment | added | Niven Zhao | @MohammadGhomi thank you for your support,I am trying to solve it since this Friday, but haven't got something important yet. It seems there's a big gap between curves on a ellipsoid and on a sphere. That's not just a question can be easily solved by using affine transformation. | |
| Mar 24, 2018 at 23:44 | comment | added | j.c. | I'm voting to reopen per @MohammadGhomi's comment. | |
| Mar 24, 2018 at 20:21 | history | edited | Johannes Hahn | CC BY-SA 3.0 |
TeXified question a bit
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| Mar 24, 2018 at 19:41 | review | Reopen votes | |||
| Mar 26, 2018 at 4:09 | |||||
| S Mar 24, 2018 at 19:24 | history | suggested | Mohammad Ghomi | CC BY-SA 3.0 |
improved formatting
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| Mar 24, 2018 at 16:19 | review | Suggested edits | |||
| S Mar 24, 2018 at 19:24 | |||||
| Mar 24, 2018 at 16:13 | comment | added | Mohammad Ghomi | This is a very natural and interesting question, and definitely constitutes research level mathematics in my opinion. I do not see why it is put on hold at all. Anyone who thinks the answer is trivial should try to solve it. You would be surprised. There are very few curves for which an intrinsic characterization is known, and it would be great if there are some reasonable equations which would characterize curves on an ellipsoid. | |
| Mar 23, 2018 at 16:47 | history | closed |
Will Jagy Igor Rivin Peter Michor abx Loïc Teyssier |
Not suitable for this site | |
| Mar 23, 2018 at 9:00 | comment | added | Robert Bryant | I suggest that you first derive the conditions for a plane curve to be an ellipse. That will give you a start. The best way to do this (and to do the higher dimensional case) is to use the moving frame for affine curves to derive the condition for lying on a hyperquadric, and then, use the Euclidean moving frame to compute the affine moving frame. This will give you the conditions you want. A good recent source would be J. Clelland's book "From Frenet to Cartan: The Method of Moving Frames". | |
| Mar 23, 2018 at 8:12 | comment | added | Peter Michor | Use a diagonal linear (or affine) transformation to translate the conditions for a sphere into that of an ellipsoid. | |
| Mar 23, 2018 at 4:20 | history | edited | Martin Sleziak | CC BY-SA 3.0 |
removed deprecated (geometry) tag - see the tag info: http://mathoverflow.net/tags/geometry/info; if there are some other geometry-related tags which are suitable, please use some of them instead
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| Mar 23, 2018 at 3:26 | comment | added | Niven Zhao | @NateEldredge I'll use my computer to recompost this question with LaTeX code later this day | |
| Mar 23, 2018 at 3:18 | history | edited | Niven Zhao | CC BY-SA 3.0 |
deleted 2 characters in body
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| Mar 23, 2018 at 3:15 | comment | added | Niven Zhao | @NateEldredge sorry,but I'm posting by my smart phone .It seems the app doesn't support type LaTex codes | |
| Mar 23, 2018 at 3:04 | comment | added | Nate Eldredge | By the way, you can use LaTeX-style math formatting. | |
| Mar 23, 2018 at 3:03 | history | edited | Niven Zhao | CC BY-SA 3.0 |
added 200 characters in body
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| Mar 23, 2018 at 2:57 | comment | added | Niven Zhao | @NateEldredge thank you for your advice | |
| Mar 23, 2018 at 2:46 | comment | added | Nate Eldredge | Then it would be a good idea if you didn't state it like a homework problem, and explained the context and any known progress or partial results. | |
| Mar 23, 2018 at 2:22 | comment | added | Niven Zhao | @NateEldredge this is not my homework.Since I've known the result about a curve lie on a sphere, I just want to know if there is a similar result on a ellipsoid. | |
| Mar 23, 2018 at 1:50 | review | Close votes | |||
| Mar 23, 2018 at 16:49 | |||||
| Mar 23, 2018 at 1:34 | review | First posts | |||
| Mar 23, 2018 at 6:40 | |||||
| Mar 23, 2018 at 1:32 | history | asked | Niven Zhao | CC BY-SA 3.0 |