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MathGroup Archive 2005

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Converting between Spherical and Cartesian coordinates

  • To: mathgroup at smc.vnet.net
  • Subject: [mg53296] Converting between Spherical and Cartesian coordinates
  • From: Alain Cochard <alain at geophysik.uni-muenchen.de>
  • Date: Wed, 5 Jan 2005 01:21:09 -0500 (EST)
  • Reply-to: alain at geophysik.uni-muenchen.de
  • Sender: owner-wri-mathgroup at wolfram.com

I use spherical coordinates:

     In[1]:= << Calculus`VectorAnalysis` ; 

     In[2]:= SetCoordinates[Spherical]

     Out[2]= Spherical[Rr, Ttheta, Pphi]

and I convert:

     In[3]:= CoordinatesFromCartesian[{x,y,z}]

		    2    2    2                  z
     Out[3]= {Sqrt[x  + y  + z ], ArcCos[------------------], ArcTan[x, y]}
					       2    2    2
					 Sqrt[x  + y  + z ]

Now, if I use the expressions for x, y, and z, I expect to get back to
Rr, Ttheta, Pphi, but:

     In[4]:= FullSimplify[%3/.{
			  x->Rr Sin[Ttheta]Cos[Pphi],
			  y->Rr Sin[Ttheta]Sin[Pphi],
			  z->Rr Cos[Ttheta]}, 
		  Rr>=0 && 0 <= Ttheta <= Pi && -Pi < Pphi <= Pi]

     Out[4]= {Rr, Ttheta, ArcTan[Rr Cos[Pphi] Sin[Ttheta], 

     >     Rr Sin[Pphi] Sin[Ttheta]]}


Even if I remove the equalities from the assumptions, I am not quite
there: 

     In[5]:= FullSimplify[%3/.{
			  x->Rr Sin[Ttheta]Cos[Pphi],
			  y->Rr Sin[Ttheta]Sin[Pphi],
			  z->Rr Cos[Ttheta]}, 
		  Rr>0 && 0 < Ttheta < Pi && -Pi < Pphi < Pi]

     Out[5]= {Rr, Ttheta, ArcTan[Cos[Pphi], Sin[Pphi]]}

But I would have thought that for a given Pphi in (-Pi,Pi), there is a
unique value for ArcTan[Cos[Pphi], Sin[Pphi]], which is... Pphi.  And
this seems to me consistent with the fact that a given (single) point
is unique, or, in other words, that to a given (perhaps
not-too-specific) {x,y,z} corresponds a unique {Rr, Ttheta, Pphi}, and
vice versa.

What am I missing here?

Thanks in advance,
AC



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