I'd really love it if you could give some intuition on how to derive the $x$, $y$ & $z$ coordinates from all/any of the orthogonal coordinate systems in this list, how you think about, say, bipolar coordinates if you had to re-derive the coordinate system on a desert island etc...

Apart from spherical & cylindrical I have no idea how to remember the others, these two I remember because I can think of the picture & re-derive how to express $x$, $y$ & $z$ in terms of $r$, $θ$ & $z$ etc... but the others are completely crazy geometrically & I have no intuition on them, & I need to at least learn how to represent $x$, $y$ & $z$ in terms of each system (so I can get grad, div, curl etc...) along with intuition on when to use them.

One interesting example of what I'm hoping for is with toroidal coordinates whose wiki is incomprehensible yet apparently there is an insanely simply way (page 114, also in this link) to derive a weak version of this coordinate system via a picture, any intuition on the rest of them?

The most important example, however, is confocal ellipsoidal coordinates (the easiest mathematical derivation of which, is given here). This aspect of the question has also been asked in this post, and it sets the tone for this question. The closest thing to a geometric interpretation for this system is given in Hobson, again though, no picture, merely a special case of what I asked for in my post and the pictures in my post as also special cases of the general case. Mathematically ellipsoidal coordinates are the most important because one can derive all the other coordinates from these by simple substitutions (a link to these substitutions in Morse and Feshbach is in the "in this post" link), ideally the goal will be to give geometric interpretations for all these substitutions as well (so we'd have two ways of getting all the coordinate systems!).

So: a) Draw a picture, b) Draw the projections determining the coordinates, analogous to this:

enter image description here

  • $\begingroup$ So what your looking for is like an abstraction of orthogonal trajectories in 2d to a 3d version? For intuition, I would say sosmath.com/diffeq/first/orthogonal/orthogonal.html . For the 3d case, I think something along these lines is what you're looking for: math.stackexchange.com/questions/588701/… $\endgroup$ – cnick Jun 9 '14 at 21:28
  • $\begingroup$ No, not that - I'm literally asking someone to copy the picture in the post for all 11 3D separable orthogonal systems, making sure to include how one derives the $x$, $y$ and $z$ expressions in terms of the new variables in each case, including some intuition about how one derives these, and hopefully also explaining how to obtain these systems from the substitutions in Morse and Feshbach. Even better would be pictures/intuition including scale factors! $\endgroup$ – bolbteppa Jun 9 '14 at 21:55
  • $\begingroup$ So here is another example: cylindrical coordinates: 3.bp.blogspot.com/_ojZ4hOEwPGo/Sw5J5z7P1II/AAAAAAAAAFs/… Almost the same picture as the one given. Note literally every resource on all the other coordinate systems only gives ugly horrendous and un-understandable representations of the coordinate systems by drawing constant surfaces which makes little sense, and never include the projections :( $\endgroup$ – bolbteppa Jun 10 '14 at 0:46
  • $\begingroup$ The simplified version of toroidal coordinates tempest.das.ucdavis.edu/pdg/intpol/code.html & the basis vectors imagebank.osa.org/… (Doesn't explain the general crazy version of toroidal coordinates given here en.wikipedia.org/wiki/Toroidal_coordinates) $\endgroup$ – bolbteppa Jun 10 '14 at 0:58
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    $\begingroup$ Hrm, too bad we aren't getting more solutions here... $\endgroup$ – rschwieb Jun 16 '14 at 12:38

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