Talk:Oblate spheroidal coordinates

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[edit] 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)

[edit] 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 = aξη

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)