Scale Factor of Oblate Spheroidal Coordinate System

Question

From WM 11.1 CoordinateChartsData returns the following scale factor:

 param = CoordinateChartData[{{"OblateSpheroidal", {\[FormalA]}}, 
    "Euclidean", 3}, "StandardCoordinateNames"];   
CoordinateChartData[{{"OblateSpheroidal", {a}}, "Euclidean", 3}, 
       "ScaleFactors", param] /. x_String :> ToExpression[x] // Column

Where a is any constant and is the elliptical focus point. The code returns the following result:

\begin{array}{l} \frac{a \sqrt{\cos (2 \eta )+\cosh (2 \xi )}}{\sqrt{2}} \\ \frac{a \sqrt{\cos (2 \eta )+\cosh (2 \xi )}}{\sqrt{2}} \\ a \sin (\eta ) \cosh (\xi ) \\ \end{array}

But this result is different from what was published in the MathWorld Encyclopedia

Also contradicts what is published at Field Theory Handbook DOI: 10.1007/978-3-642-83243-7

Can anyone comment which is correct and if there might be a bug?

Answer 1

Depends what you mean by "correct", but I don't think there is a bug. There are always multiple conventions which are chosen for whatever reason, so you need to be careful when making comparisons. We tended to follow Neutsch, but I think there were a couple of places we decided someone else did better (I don't immediately recall in which category OblateSpheroidal was). And worst of all, sometimes different sources will use the same name for different coordinate systems.

The first thing to check is the coordinate ranges used, which differ between CoordinateChartData and MathWorld:

In[38]:= CoordinateChartData[{{"OblateSpheroidal", {a}}, "Euclidean", 3},
            "CoordinateRangeAssumptions", {ξ, η, ϕ}]

Out[38]= ξ > 0 && 0 < η < π && -π < ϕ <= π

The change in ϕ is fairly minor. It doesn't affect most formulas, but it is consistent with our branch cuts in ArcTan and thus leads to better simplification. The change to η is more significant, going from latitude to colatitude. This ensures that the coordinate system is right-handed, which is pretty important if you're going to be doing vector calculus and not just solving Laplace's equation. MathWorld's Oblate Spheroidal is left-handed. I don't have a copy of Moon & Spencer handy, but I believe that it is the same as MathWorld's. I specifically rejected that convention when working on CoordinateChartData because it leads to a left-handed coordinate system.

Finally, if you make the substitution η -> π/2 - η and compare these scale factors with MathWorld's, you should find that they are the same:

In[40]:= FullSimplify[a*Sqrt[-Cos[2*η] + Cosh[2*ξ]]/Sqrt[2] ==
                      a*Sqrt[Sinh[ξ]^2 + Sin[η]^2]]
Out[40]= True