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[?] Get expected transformation from OblateSpheroidal to Cartesian?

Jose Calderon

Jose Calderon, University of Puerto Rico

Posted 8 years ago

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?

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?

POSTED BY: Jose Calderon

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Sander Huisman

Sander Huisman, University of Twente

Posted 8 years ago

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

POSTED BY: Sander Huisman

Sander Huisman

Sander Huisman, University of Twente

Posted 8 years ago

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

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

POSTED BY: Sander Huisman

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