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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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