Talk:Oblate spheroidal coordinates

Downgrade the sigma-tau-phi version

I'm in favor of downgrading the sigma-tau-phi version of the coordinates. I've never seen them used and I don't think they are as useful as coordinates which are one-to-one with each point in space. I think we should just mention them and note that they are degenerate. PAR (talk) 23:39, 3 February 2008 (UTC)

I've no strong feelings about the exposition, but I'd definitely like to keep the image showing the degeneracy. It's a precursor of the other images of degenerate ellipsoidal coordinates that I intend to make. The σ–τ coordinates are found in the Korn&Korn reference, and would presumably be useful in symmetrical situations, no? The formulae for the differential operators seem to be a little simpler, perhaps that's why they're used? I'll go to the library and try to find other instances where they're used in the scientific literature. If we can find such instances, I'd be in favor of keeping them, but maybe putting them last, after your analogous but non-degenerate ξ–η coordinates? Willow (talk) 01:26, 4 February 2008 (UTC)

PS. How do you like the 3D images so far? Do you have any suggestions?

That sounds good, for the sigma-tau coordinates. I think the 3D images are EXCELLENT, but they show up a bit dark on my machine. I don't know whether that is me or the images. How did you generate them? PAR (talk) 16:27, 4 February 2008 (UTC)

I'm really glad that you like them. I made them with Blender, which I started learning late at night at Christmas time. It's amazingly powerful. I find the images a little dark, too, but I'm just beginning to learn how to use the program, so I hope to learn eventually how to fix it. I wanted the isosurfaces to look like blown glass, but without being too shiny. Willow (talk) 19:25, 4 February 2008 (UTC)

I think the scale factors (and thus possibly the Laplacian) are wrong for the sigma-tau version. They should be

$h_{\sigma }=a{\sqrt {\frac {\sigma ^{2}-\tau ^{2}}{\sigma ^{2}-1}}}$

$h_{\tau }=a{\sqrt {\frac {\sigma ^{2}-\tau ^{2}}{1-\tau ^{2}}}}$

These I got from the general formula:

$h_{\sigma }^{2}=\left({\frac {\partial x}{\partial \sigma }}\right)^{2}+\left({\frac {\partial y}{\partial \sigma }}\right)^{2}+\left({\frac {\partial z}{\partial \sigma }}\right)^{2}$

I agree that these scale factors (and maybe the Laplacian too) are wrong for this version of the coordinates and should be those given directly above. I derived them as the eigenvalues of the tensor

$g_{ij}=\sum _{k}{\frac {\partial r_{k}}{\partial q_{i}}}{\frac {\partial r_{k}}{\partial q_{j}}}$

where the $\{r_{i}\}=\{x,y,z\}$ and the $\{q_{i}\}=\{\sigma ,\tau ,\phi \}$. Furthermore, the scale factors derived on the actual page can be derived given an incorrect sign in the $\sigma$ term of $z$:

$z^{2}=a^{2}\left(\sigma ^{2}+1\right)\left(1-\tau ^{2}\right).$

This is likely the mistake. —Preceding unsigned comment added by 67.85.252.69 (talk) 00:58, 25 February 2011 (UTC)

Different conventions

Hey PAR,

I went to the library and dug up a few books that had stuff about these coordinates. Morse and Feshbach have really pretty stereo images, have you seen them? Basically, they use your definitions

$x=a\cos \phi {\sqrt {\left(\xi ^{2}+1\right)\left(1-\eta ^{2}\right)}}$

$y=a\sin \phi {\sqrt {\left(\xi ^{2}+1\right)\left(1-\eta ^{2}\right)}}$

$z=a\xi \eta$

where ξ = sinh μ and η = sin ν. They express it slightly differently, though, using (ξ₁, ξ₂, ξ₃) defined as

$\xi _{1}\equiv =a\xi ,\xi _{2}\equiv \eta ,\xi _{3}=\cos \phi$

I found the same definitions in the Handbook of Integration by Daniel Zwillinger, except using (u₁, u₂, u₃). It seems that most people use the colatitude θ instead of the latitude ν; I don't know how I got the impression that the latitude was more commonly used? Do you think we should switch from ν to θ? I rather like your 2D plot and would be sorry to see it go. Willow (talk) 19:25, 4 February 2008 (UTC)

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I think the sum giving V=sigma(n=0 to infinity)sigma(m=0 to infinity) could be replaced by V=sigma(n=0 to infinity)sigma(m=0 to n). —Preceding unsigned comment added by Tournesol007 (talk • contribs) 20:01, 1 November 2010 (UTC)

Wrong expressions of Laplacian

The expressions of Laplacian seems to be wrong in this article, i.e. not compatible with the expression derived from the article "Orthogonal Coordinates" which seems correct upon setting the metric factors appropriately. The wrong feature is in the latitudinal coordinate component, which should contain a factor inside the first derivative operator (the factor has been put out the derivative as if it did not depend on the coordinate). — Preceding unsigned comment added by PBenard (talk • contribs) 20:11, 18 May 2012 (UTC)

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I added a "+" sign in the Laplacian's last term between the squared hyperbloic cosine and the squared cosine. This was an error in addition to the possible error mentioned above.Smokey Martini 19:43, 1 July 2012 (UTC) — Preceding unsigned comment added by Smokey martini (talk • contribs)

This article defines a left-handed coordinate system

Why is this coordinate system defined in a left-handed way? It's not wrong, but it is odd. In analogy to spherical coordinates, we can make the identification $(\mu ,\nu ,\phi )\rightarrow (r,\theta ,\phi )$. The latter is clearly left-handed if we take $\theta$ to be a latitude (increasing from "south" to "north." Since we are here defining $\nu \in \{-\pi /2,\pi /2\}$ (like a latitude, rather than a colatitude), I would expect the coordinates to be defined as $(\phi ,\nu ,\mu )$ in analogy to the right-handed latitudinal spherical coordinates $(\phi ,\theta ,r)$.