Timeline for Differential rotations in Chebyshev net
Current License: CC BY-SA 4.0
37 events
| when toggle format | what | by | license | comment | |
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| Feb 16, 2019 at 13:12 | history | edited | YCor | CC BY-SA 4.0 |
fixed typos
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| Feb 16, 2019 at 6:35 | history | edited | Narasimham | CC BY-SA 4.0 |
added isometric differential rotation product identity
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| Nov 22, 2017 at 8:37 | history | edited | YCor | CC BY-SA 3.0 |
corrected misspelled Chebyshev
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| Jul 23, 2016 at 15:31 | comment | added | Ben McKay | As Robert Bryant pointed out, if your surface has metric $ds^2$ (which would coincide with the conventional notation) and $K$ is a negative constant, then the expression $d\phi_1^2 + d\phi_2^2=-K \, ds^2$ ensures that the metric is a constant multiple of a flat metric $ds^2=(-1/K)(d\phi_1^2+d\phi_2^2)$, as Robert Bryant pointed out, since it is a multple of the flat Euclidean metric in the coordinates $\phi_1, \phi_2$. You seem to be ignoring this problem. I don't know what "asymptotic differential rhombic element corners" are; please give a reference to define $\phi_1, \phi_2$. | |
| Jul 22, 2016 at 18:49 | comment | added | Narasimham | @ Todd Trimble The "in the large" relevance is necessary, I ought to have mentioned it in the first place. | |
| Jul 22, 2016 at 18:46 | history | edited | Narasimham | CC BY-SA 3.0 |
The cosh form (in the large) is given for context.
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| Jul 20, 2016 at 19:50 | comment | added | Narasimham | This can happen as the arc is straight. With $ R_n = 1/k_n= \infty $ the metric need not necessarily be the only flat metric. The asymptotic line of constant negative $K$ can have an admissible non-euclidean claim for generalization of the metric. | |
| Jul 20, 2016 at 18:19 | comment | added | Narasimham | And now we have differential arc lengths $ a \, d\phi_1,a \, d\phi_2 $ obeying the Pythagoras theorem with hypotenuse lying along asymptotic line and sides along direction of extreme normal curvatures. | |
| Jul 20, 2016 at 18:11 | comment | added | Narasimham | For asymptotic lines all points are inflection points. Asymptotic curves have constant geodesic torsion $\frac{1}{a},$ being straight along this direction, Tangent plane is same as (coincides with) osculating plane ... as is known. | |
| Jul 18, 2016 at 18:56 | comment | added | Narasimham | To all... The requested proposition has been self answered in the derivation of equations (7,8) confirming that these rotations in fact are the rotational parameters that characterize non-Euclidean hyperbolic geometry metric after division by an invariant $ \sqrt {-K} $ .. a view for which an answer is awaited. | |
| Jul 17, 2016 at 21:41 | comment | added | Todd Trimble | Please be aware that each edit bumps the question to the front of the active questions list, at the expense of other questions which are vying for attention. Better is to do only a few edits to correct many infelicities within the same edit, rather than many edits to do one or two at a time. | |
| Jul 17, 2016 at 20:22 | history | edited | Narasimham | CC BY-SA 3.0 |
showing angle between geodesics in picture and minor grammar..
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| Jul 17, 2016 at 9:38 | history | edited | Narasimham | CC BY-SA 3.0 |
very minor grammar and tag changes
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| Jul 17, 2016 at 5:50 | history | edited | Narasimham | CC BY-SA 3.0 |
boxed the new arrival (in the OP's view ! ) of hyperbolic geodesic Pythagoras Theorem and my view.
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| Jul 17, 2016 at 5:17 | history | edited | Narasimham | CC BY-SA 3.0 |
Equation (5) boxed
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| Jul 16, 2016 at 21:30 | comment | added | Narasimham | Replacing dimensionless lengths by angles this way is fundamental to non-Euclidean geometries as it appears to me.. generating lengths via small angular arguments and then compensatingly back again with a dimensional invariant , as in: $ a\, \sinh\, s/a \approx s .$ Shallow first order approximations result when $ s<< a$ working out to distances/ arc lengths in Euclidean geometry. | |
| Jul 16, 2016 at 19:03 | history | edited | Narasimham | CC BY-SA 3.0 |
added 8 characters in body
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| Jul 16, 2016 at 18:53 | comment | added | Narasimham | @ Robert Bryant At the end is included a short derivation for the apparently " flat metric".I wish to know if there are more fundamental ways to prove it...however.... | |
| Jul 16, 2016 at 18:44 | history | edited | Narasimham | CC BY-SA 3.0 |
derivation for new hyperbolic metric
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| Jul 16, 2016 at 17:07 | history | edited | Narasimham | CC BY-SA 3.0 |
max/min instead of principal
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| Jul 16, 2016 at 16:52 | history | edited | Narasimham | CC BY-SA 3.0 |
two solutions briefly mentioned between (4) and (5)
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| Jul 16, 2016 at 15:17 | comment | added | Narasimham | These rotations are calculated and culminate in correct equations (4). Does taking as induced metric help? | |
| Jul 16, 2016 at 14:59 | history | edited | Narasimham | CC BY-SA 3.0 |
deleted 1 character in body
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| Jul 16, 2016 at 12:30 | history | edited | Narasimham | CC BY-SA 3.0 |
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| Jul 16, 2016 at 12:09 | history | edited | Narasimham | CC BY-SA 3.0 |
added 15 characters in body
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| Jul 16, 2016 at 11:42 | history | edited | Narasimham | CC BY-SA 3.0 |
corrected typo in (3) and defined $\psi$
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| Jul 16, 2016 at 11:36 | history | edited | Narasimham | CC BY-SA 3.0 |
corrected typo in (3)
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| Jul 16, 2016 at 9:54 | history | edited | Narasimham | CC BY-SA 3.0 |
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| Jul 16, 2016 at 9:48 | history | edited | Narasimham | CC BY-SA 3.0 |
added 2037 characters in body
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| Jul 16, 2016 at 9:13 | comment | added | Robert Bryant | Something is wrong with your equation. The metric on the left, $d\phi_1^2+d\phi_2^2$, is obviously flat, but the metric on the right, $-K ds^2$, has constant negative Gauss curvature $-1$. Or do you not mean $ds^2$ to be the induced metric? | |
| Jul 15, 2016 at 17:06 | history | edited | Narasimham | CC BY-SA 3.0 |
added 65 characters in body
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| Jul 15, 2016 at 15:04 | history | edited | Narasimham | CC BY-SA 3.0 |
including parameters.
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| Jul 15, 2016 at 12:29 | history | edited | Narasimham | CC BY-SA 3.0 |
Normals drawn better
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| Jul 15, 2016 at 10:44 | history | edited | Narasimham | CC BY-SA 3.0 |
differential rhombus between hyperbolic geodesic parallels added
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| Jul 14, 2016 at 23:39 | history | edited | Narasimham | CC BY-SA 3.0 |
added 13 characters in body
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| Jul 14, 2016 at 23:24 | history | edited | Narasimham | CC BY-SA 3.0 |
spelling minor
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| Jul 14, 2016 at 22:48 | history | asked | Narasimham | CC BY-SA 3.0 |