Timeline for Geodesics on the twisted pseudosphere (Dini's surface)
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Apr 15, 2019 at 21:01 | comment | added | Narasimham | @RobertBryant What may be the pure invariant twisting parameter that relates to each degree(or magnitude) of twist ? I suppose it is a natural question to ask. (Related to my earlier question here ... in search of a twisting invariant). The corresponding question for pure bending is well known. | |
| Nov 27, 2013 at 14:06 | comment | added | j.c. | I think you get the right branch (even in Mathematica) if you integrate $f'$ from $\theta$ up to $\pi/2-t$. When I do this (here $h=\theta$), I get the expression (in Mathematica notation) f[h_]:=1/2 (Log[2]-2 Log[Sqrt[2] Cos[h] + Sqrt[Cos[2 t] + Cos[2 h]]] + 2 Log[Sin[t]] + Abs[Cos[t]] (2 ArcTanh[(Sqrt[2] Abs[Cos[t]] Cos[h])/Sqrt[Cos[2 t] + Cos[2 h]]] - Log[-Sec[t] Sqrt[-Sin[2 t]]] + Log[Sec[t] Sqrt[-Sin[2 t]]])) and this is a real function for $0<h<\pi/2-t$. This $f$ diverges at 0 (c.f. the "unreachable" geodesic on the curve $\theta=0$) and vanishes at $\pi/2-t$ (the cusp). | |
| Nov 27, 2013 at 4:45 | history | edited | Robert Bryant | CC BY-SA 3.0 |
removed the comment about f not being elementary, since it was wrong
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| Nov 27, 2013 at 4:29 | comment | added | Robert Bryant | This must be a problem with choosing the right branch of some analytic continuation. Obviously, $f'$ is real in the range in question, so its integral will be, too. On the other hand, I see now, that I was wrong when I said that $f$ is not elementary. I made an incorrect change of variables when I attempted to integrate the given $f'$ and wound up with an elliptic integral. I'll fix this. | |
| Nov 27, 2013 at 1:40 | comment | added | Joseph O'Rourke | When I (mindlessly) integrate $f'(\theta)$, I obtain $\sqrt{\cos ^2(t)} \tanh ^{-1}\left(\frac{\cos (\theta ) \sqrt{\cos (2 t)+1}}{\sqrt{\cos (2 \theta )+\cos (2 t)}}\right)+\log \left(\sqrt{2} \cos (\theta )+\sqrt{\cos (2 \theta )+\cos (2 t)}\right)$, which yields imaginary values for $\theta$ less than $\pi/2$... | |
| Nov 26, 2013 at 18:29 | comment | added | Robert Bryant | You are welcome, Joseph; It was a fun exercise. The geodesics you mention are, in the $xy$-coordinates, given by $y = c$ where $c>0$ is a constant. In the $\rho\theta$-coordinates, this is $\rho=c/(\sin\theta)$, and the parametrization that goes down to $z=-\infty$ is when $\theta$ goes down from $\tfrac12\pi{-}t>0$ to $0$. Substituting this into the above formulae gives you the curve in the $uv$-coordinates, then, of course, you substitute into the $xyz$-parametrization to get the curve on Dini's surface. The $xyz$-formulae may simplify, but they won't be elementary because $f$ isn't. | |
| Nov 26, 2013 at 11:52 | comment | added | Joseph O'Rourke | Thank you, Robert! At some point in the future, I'd like to compute one of the geodesics curling down to $z=-\infty$, just for the aesthetic pleasure. | |
| Nov 26, 2013 at 10:44 | vote | accept | Joseph O'Rourke | ||
| Nov 26, 2013 at 4:35 | comment | added | Robert Bryant | @Joseph: Well, that wasn't what I had in mind when I used the word, but, OK. If you like words, might I suggest that Dini's surface, rather than being described as a 'twisted pseudosphere', might be more aptly described as 'unfurled'. | |
| Nov 26, 2013 at 2:13 | comment | added | Joseph O'Rourke | I am happy to learn the word nappe!: "1. a sheet of rock that has moved sideways over neighboring strata as a result of an overthrust or folding. 2. A sheet of water flowing over a dam or similar structure." | |
| Nov 25, 2013 at 21:53 | history | edited | Robert Bryant | CC BY-SA 3.0 |
Noticed a nicer way to express the coordinate change
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| Nov 25, 2013 at 21:31 | history | edited | Robert Bryant | CC BY-SA 3.0 |
Added the desired change of coordinates
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| Nov 25, 2013 at 14:19 | history | edited | Robert Bryant | CC BY-SA 3.0 |
fixed grammar an typos
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| Nov 25, 2013 at 14:09 | history | answered | Robert Bryant | CC BY-SA 3.0 |