       Re: Converting between Spherical and Cartesian coordinates

• To: mathgroup at smc.vnet.net
• Subject: [mg53372] Re: Converting between Spherical and Cartesian coordinates
• From: Alain Cochard <alain at geophysik.uni-muenchen.de>
• Date: Sat, 8 Jan 2005 23:02:32 -0500 (EST)
• References: <16858.22033.555637.91757@localhost.localdomain>
• Sender: owner-wri-mathgroup at wolfram.com

```Alain Cochard writes:

> I use spherical coordinates:
>
>      In:= << Calculus`VectorAnalysis` ;
>
>      In:= SetCoordinates[Spherical]
>
>      Out= Spherical[Rr, Ttheta, Pphi]
>
> and I convert:
>
>      In:= CoordinatesFromCartesian[{x,y,z}]
>
> 		    2    2    2                  z
>      Out= {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:= 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= {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:= 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= {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?

In fact I had not properly searched the archives.  While looking for
something else I found the solution, namely by use of the PowerExpand
transformation function:

In:= 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,
TransformationFunctions -> {Automatic,  PowerExpand}]

Out= {Rr, Ttheta, Pphi}

AC

```

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