[?] Get expected transformation from OblateSpheroidal to Cartesian?

Questions
,
1

I am trying to plot a line in Oblate Spheroidal Coordinate system (OS). I want to seek the equation in the OS from the code

CoordinateTransformData[
 "Spherical" -> "Cartesian", "Mapping", {r, \[Theta], \[CurlyPhi]}]
CoordinateTransformData[
 "Cartesian" -> "Spherical", "Mapping", {x, y, z}]
CoordinateTransformData["Cartesian" -> "Spherical", 
  "Mapping", {r Cos[\[CurlyPhi]] Sin[\[Theta]], 
   r Sin[\[Theta]] Sin[\[CurlyPhi]], r Cos[\[Theta]]}] // FullSimplify

But the result I am getting is not the same

FullSimplify[{Sqrt[r^2], 
   ArcTan[r Cos[\[Theta]], Sqrt[r^2 Sin[\[Theta]]^2]], 
   ArcTan[r Cos[\[CurlyPhi]] Sin[\[Theta]], 
    r Sin[\[Theta]] Sin[\[CurlyPhi]]]} == {r, \[Theta], \[CurlyPhi]}]



Out[62]= {Sqrt[x^2+y^2+z^2],ArcTan[z,Sqrt[x^2+y^2]],ArcTan[x,y]}
Out[63]= {Sqrt[r^2],ArcTan[r Cos[\[Theta]],Sqrt[r^2 Sin[\[Theta]]^2]],ArcTan[r Cos[\[CurlyPhi]] Sin[\[Theta]],r Sin[\[Theta]] Sin[\[CurlyPhi]]]}
In[65]:= FullSimplify[{Sqrt[r^2],ArcTan[r Cos[\[Theta]],Sqrt[r^2 Sin[\[Theta]]^2]],ArcTan[r Cos[\[CurlyPhi]] Sin[\[Theta]],r Sin[\[Theta]] Sin[\[CurlyPhi]]]}=={r,\[Theta],\[CurlyPhi]}]
Out[65]= {Sqrt[r^2],ArcTan[r Cos[\[Theta]],Sqrt[r^2 Sin[\[Theta]]^2]],ArcTan[r Cos[\[CurlyPhi]] Sin[\[Theta]],r Sin[\[Theta]] Sin[\[CurlyPhi]]]}=={r,\[Theta],\[CurlyPhi]}

Not getting the transformation from OS back to its cartesian coordiantes. Why is this?

2

You need to include some assumptions, r being positive, theta being real, et cetera:

Something like:

FullSimplify[{Sqrt[r^2], 
   ArcTan[r Cos[\[Theta]], Sqrt[r^2 Sin[\[Theta]]^2]], 
   ArcTan[r Cos[\[CurlyPhi]] Sin[\[Theta]], 
    r Sin[\[Theta]] Sin[\[CurlyPhi]]]} == {r, \[Theta], \[CurlyPhi]}, 
 r > 0 \[And] 
  0 < \[Theta] < Pi \[And] \[Theta] \[Element] Reals \[And] 
  0 < \[CurlyPhi] < Pi \[And] \[CurlyPhi] \[Element] Reals]

returns:

True
3

Moreover, not all these mappings are ‘unique’, so going from spherical to cartesian and back to spherical does not always give the same coordinates. Examples include the poles and the center for which for r=0