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>
- Reply-to: alain at geophysik.uni-muenchen.de
- Sender: owner-wri-mathgroup at wolfram.com
Alain Cochard writes:
> 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?
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[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,
TransformationFunctions -> {Automatic, PowerExpand}]
Out[5]= {Rr, Ttheta, Pphi}
AC