Editing Apps/Ostriker64 (section)

==Coordinate System==

===Basics===
In &sect;IIa of [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II], Ostriker defines a set of orthogonal coordinates, <math>~(r,\phi,\theta)</math>, that is related to the traditional Cartesian coordinate system, <math>~(x,y,z)</math>, via the relations,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(R+r\cos\phi)\cos\theta \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~y</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(R+r\cos\phi)\sin\theta \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r\sin\phi \, .</math>
  </td>
</tr>
</table>
As Ostriker states, <font color="darkgreen">"The coordinate <math>~r</math> is the distance from a reference circle of radius <math>~R</math> (later chosen to be the major radius of the ring) &hellip;"</font>  The angle, <math>~\theta</math>, plays the role of the azimuthal angle, as is familiar in both cylindrical and spherical coordinates, while, here, <math>~\phi</math> is a meridional-plane polar angle measured counterclockwise from the equatorial plane.  For axisymmetric systems, there will be no dependence on the azimuthal angle, so the pair of relevant coordinates in the meridional plane are, 

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\varpi \equiv (x^2+y^2)^{1 / 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~R+r\cos\phi \, ,</math>
  </td>
  <td align="center">&nbsp; &nbsp; and, &nbsp; &nbsp;
  <td align="right">
<math>~z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r\sin\phi \, .</math>
  </td>
</tr>
</table>

<div align="center" id="THH12Figure4">
<table border="1" cellpadding="8">

<tr><td align="center">
Figure 1 extracted without modification from p. 1077 of [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O J. P. Ostriker (1964; Paper II)]<p></p>
"''The Equilibrium of Self-Gravitating Rings''"<p></p>
ApJ, vol. 140, pp. 1067-1087 &copy; American Astronomical Society
</td>
</tr>

<tr>
<td align="center">
[[File:Ostriker64PaperIIFig1.png|600px|Figure 1 from Ostriker (1964) Paper II]]
</td>
</tr>
</table>
</div>
For later reference, we note that (see eq. 3 of Paper II) the corresponding line element is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\delta s^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\delta r^2 + r^2 \delta\phi^2 + (R+r\cos\phi)^2\delta\theta^2 \, ,
</math>
  </td>
</tr>
</table>
which means that the relevant scale factors for the adopted coordinate system, <math>~(r,\phi,\theta)</math>, are
<div align="center">
<math>~h_1 = 1 \, ,</math> &nbsp; &nbsp; &nbsp; <math>~h_2 = r \, ,</math> &nbsp; &nbsp; &nbsp; <math>~h_3 = (R+r\cos\phi) \, ,</math>
</div>
and the relevant differential volume element is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~d^3 x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~h_1 h_2 h_3 dr d\phi d\theta
= r(R+r\cos\phi) dr d\phi d\theta\, .
</math>
  </td>
</tr>
</table>

===Relationship to Toroidal Coordinate===
Referring back to our separate discussion of the [[2DStructure/ToroidalGreenFunction#Basic_Elements_of_a_Toroidal_Coordinate_System|basic elements of a toroidal coordinate system]], we know that, the meridional-plane toroidal coordinates <math>~(\eta,\theta)</math> are related to traditional meridional-plane cylindrical coordinate pair <math>~(\varpi,z)</math> via the expressions,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\varpi}{R}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\sinh\eta}{\cosh\eta - \cos\theta} \, ,</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
  <td align="right">
<math>~\frac{z}{R}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\sin\theta}{\cosh\eta - \cos\theta} \, ,</math>
  </td>
</tr>
</table>

assuming that the cylindrical-coordinate location of the ''anchor ring'' is <math>~(\varpi,z) = (R,0)</math>.  Let's determine how to transform between these two sets of coordinate pairs.  

====Independent Exploration====
First, eliminating reference to Ostriker's "polar angle" <math>~\phi</math>, we see that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{r^2}{R^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{\varpi}{R} - 1 \biggr)^2 + \biggl(\frac{z}{R}\biggr)^2</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{\sinh\eta}{\cosh\eta - \cos\theta} - 1 \biggr]^2 + \biggl[ \frac{\sin\theta}{\cosh\eta - \cos\theta} \biggr]^2</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{(\sinh\eta - \cosh\eta + \cos\theta)^2 + \sin^2\theta}{(\cosh\eta - \cos\theta)^2} \biggr] \, .</math>
  </td>
</tr>
</table>
Then, eliminating reference to Ostriker's radial coordinate <math>~r</math>, we find,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\cot\phi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\varpi/R - 1}{z/R}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\sinh\eta - \cosh\eta + \cos\theta}{\sin\theta}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\cot\theta  + \frac{\sinh\eta - \cosh\eta }{\sin\theta} \, .</math>
  </td>
</tr>
</table>
Now let's try to derive the alternate transformation.  We'll start by eliminating the "polar angle" in toroidal coordinates.
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\cosh\eta - \cos\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\sinh\eta}{\varpi/R}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \cos\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\cosh\eta - \frac{\sinh\eta}{\varpi/R} \, .</math>
  </td>
</tr>
</table>
The same relation also implies that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{z}{R}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\varpi}{R}\biggr) \frac{\sin\theta}{\sinh\eta}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \sin\theta </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{z}{R}\biggl( \frac{\varpi}{R}\biggr)^{-1} \sinh\eta \, .</math>
  </td>
</tr>
</table>
Together, then, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~1 = \sin^2\theta + \cos^2\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{z}{R}\biggl( \frac{\varpi}{R}\biggr)^{-1} \sinh\eta \biggr]^2
+ \biggl[ \cosh\eta - \frac{\sinh\eta}{\varpi/R} \biggr]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{\varpi}{R}\biggr)^{-2} \biggl[ \frac{z}{R} \cdot \sinh\eta \biggr]^2
+ \biggl(\frac{\varpi}{R}\biggr)^{-2} \biggl[ \frac{\varpi}{R}\cdot \cosh\eta - \sinh\eta \biggr]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \biggl( \frac{\varpi}{R}\biggr)^{2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{z}{R} \cdot \sinh\eta \biggr]^2
+ \biggl[ \frac{\varpi}{R}\cdot \cosh\eta - \sinh\eta \biggr]^2 \, .
</math>
  </td>
</tr>
</table>
Alternatively, in an attempt to eliminate <math>~\eta</math>, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\sinh\eta </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\varpi}{R}\biggl( \frac{z}{R}\biggr)^{-1} \sin\theta </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \cosh\eta = \biggl[ 1 + \sinh^2\eta\biggr]^{1 / 2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ 1 + \biggl(\frac{\varpi}{R}\biggr)^2 \biggl( \frac{z}{R}\biggr)^{-2} \sin^2\theta \biggr]^{1 / 2} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{z}{R}\biggr)^{-1} \biggl[ \biggl( \frac{z}{R}\biggr)^{2}  + \biggl(\frac{\varpi}{R}\biggr)^2 \sin^2\theta \biggr]^{1 / 2} \, .</math>
  </td>
</tr>
</table>
But, also,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\cosh\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{z}{R} \biggr)^{-1} \sin\theta + \cos\theta</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\biggl[ \biggl( \frac{z}{R}\biggr)^{2}  + \biggl(\frac{\varpi}{R}\biggr)^2 \sin^2\theta \biggr]^{1 / 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sin\theta +  \biggl( \frac{z}{R}\biggr) \cos\theta</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\biggl( \frac{z}{R}\biggr)^{2}  + \biggl(\frac{\varpi}{R}\biggr)^2 \sin^2\theta </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sin^2\theta + 2\biggl( \frac{z}{R}\biggr)\sin\theta \cos\theta +  \biggl( \frac{z}{R}\biggr)^2 \cos^2\theta</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\biggl( \frac{z}{R}\biggr)^{2}\biggl[1-\cos^2\theta  \biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sin^2\theta\biggl[1 -  \biggl(\frac{\varpi}{R}\biggr)^2  \biggr] + 2\biggl( \frac{z}{R}\biggr)\sin\theta \cos\theta </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~0 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sin^2\theta\biggl[1 -  \biggl(\frac{\varpi}{R}\biggr)^2  - \biggl( \frac{z}{R}\biggr)^{2}\biggr] + 2\biggl( \frac{z}{R}\biggr)\sin\theta \cos\theta </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\biggl[1 -  \biggl(\frac{\varpi}{R}\biggr)^2  - \biggl( \frac{z}{R}\biggr)^{2}\biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - 2\biggl( \frac{z}{R}\biggr) \cot\theta </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \cot\theta </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{1}{2}\biggl( \frac{z}{R}\biggr)^{-1} \biggl[1 -  \biggl(\frac{\varpi}{R}\biggr)^2  - \biggl( \frac{z}{R}\biggr)^{2}\biggr] \, .</math>
  </td>
</tr>
</table>
Now that I think about it, this is all a bit silly because from the [[2DStructure/ToroidalGreenFunction#Basic_Elements_of_a_Toroidal_Coordinate_System|basic elements of a toroidal coordinate system]] we already know how to shift from cylindrical to toroidal coordinates.

====Back to Basics====

Mapping the other direction [see equations 2.13 - 2.15 of [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] ], we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\ln\biggl(\frac{r_1}{r_2} \biggr) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\cos\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(r_1^2 + r_2^2 - 4R^2)}{2r_1 r_2} \, ,</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~r_1^2 </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~[\varpi + R]^2 + z^2 \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  <td align="right">
<math>~r_2^2 </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~[\varpi - R]^2 + z^2 \, ,</math>
  </td>
</tr>
</table>
</div>
and <math>~\theta</math> has the same sign as <math>~z</math>.  Now, given that Ostriker's <math>~(r,\phi)</math> coordinates are related to cylindrical coordinates via the expressions,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\varpi </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~R+r\cos\phi \, ,</math>
  </td>
  <td align="center">&nbsp; &nbsp; and &nbsp; &nbsp;
  <td align="right">
<math>~z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r\sin\phi \, ,</math>
  </td>
</tr>
</table>
we can write,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~r_1^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~[2R + r\cos\phi]^2 + r^2\sin^2\phi </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4R^2 + 4Rr\cos\phi + r^2 \, ;</math>
  </td>
</tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~r_2^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r^2 \, .</math>
  </td>
</tr>
</table>
Hence,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\cos\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2r[4R^2 + 4Rr\cos\phi + r^2]^{1 / 2}} \biggl[ 4Rr\cos\phi + 2r^2  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2R\cos\phi + r }{[4R^2 + 4Rr\cos\phi + r^2]^{1 / 2}} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~e^{2\eta}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{ 4R^2 + 4Rr\cos\phi + r^2 }{r^2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4\biggl( \frac{R}{r}\biggr)^2 + 4\biggl(\frac{R}{r}\biggr)\cos\phi + 1 \, . 
</math>
  </td>
</tr>
</table>

===Summary===

<table border="1" cellpadding="10" align="center" width="85%">
<tr>
  <td align="left" width="50%">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{r^2}{R^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  <td align="left">
<math>~\biggl[ \frac{(\sinh\eta - \cosh\eta + \cos\theta)^2 + \sin^2\theta}{(\cosh\eta - \cos\theta)^2} \biggr] </math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\cot\phi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\cot\theta  + \frac{\sinh\eta - \cosh\eta }{\sin\theta} </math>
  </td>
</tr>
</table>

  </td>
  <td align="left">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~e^{2\eta}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4\biggl( \frac{R}{r}\biggr)^2 + 4\biggl(\frac{R}{r}\biggr)\cos\phi + 1 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\cos\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2R\cos\phi + r }{[4R^2 + 4Rr\cos\phi + r^2]^{1 / 2}} 
</math>
  </td>
</tr>
</table>

  </td>
</tr>
</table>

===Second Attempt===
====Single Offset Circle====
Now an [[Appendix/Ramblings/ToroidalCoordinates#Off-center_Circle|off-center circle]] whose major and minor radii are, respectively, <math>~(\varpi_0,d)</math>, will be described by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~d^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(\varpi - \varpi_0)^2 + z^2 \, .
</math>
  </td>
</tr>
</table>

<span id="Dsquared">where both <math>~d</math> and <math>~\varpi_0</math> are held constant while mapping out the variation of <math>~z</math> with <math>~\varpi</math>.  If we acknowledge that, in general, <math>~\varpi_0 \ne R_\mathrm{JPO}</math>, then we know how <math>~r</math> varies with <math>~\phi</math> via the relation,</span>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~d^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ R_\mathrm{JPO} + r\cos\phi - \varpi_0\biggr]^2 + r^2\sin^2\phi 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(R_\mathrm{JPO}-\varpi_0)^2 
+ 2\biggl[ (R_\mathrm{JPO}-\varpi_0) r\cos\phi \biggr]
+r^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 0 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
r^2 + 2r\biggl[ (R_\mathrm{JPO}-\varpi_0) \cos\phi \biggr] + \biggl[(R_\mathrm{JPO}-\varpi_0)^2 - d^2\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ r </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2}\biggl\{
- 2\biggl[ (R_\mathrm{JPO}-\varpi_0) \cos\phi \biggr] \pm 
\sqrt{ 4\biggl[ (R_\mathrm{JPO}-\varpi_0) \cos\phi \biggr]^2 - 4\biggl[(R_\mathrm{JPO}-\varpi_0)^2 - d^2\biggr] }
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2}\biggl\{
2\biggl[ (\varpi_0 - R_\mathrm{JPO}) \cos\phi \biggr] \pm 
\sqrt{ 4\biggl[ (\varpi_0 - R_\mathrm{JPO}) \cos\phi \biggr]^2 - 4\biggl[(\varpi_0 - R_\mathrm{JPO})^2 - d^2\biggr] }
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ \frac{r}{ (\varpi_0 - R_\mathrm{JPO}) }</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\cos\phi  \pm 
\sqrt{ \cos^2\phi  - 1 + d^2 (\varpi_0 - R_\mathrm{JPO})^{-2} }
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\cos\phi  \pm \sqrt{ d^2 (\varpi_0 - R_\mathrm{JPO})^{-2}-\sin^2\phi }
</math>
  </td>
</tr>
</table>

In order to align this expression with the terminology (and variable labels) that we use in the context of a toroidal coordinate system, we associate the radius of the ''anchor ring'' as <math>~R_\mathrm{JPO}\leftrightarrow a</math>, and we associate the major radius of each circular torus as <math>~\varpi_0 \leftrightarrow R_0</math>.  We therefore have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{r}{ (R_0-a) }</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\cos\phi  \pm \sqrt{ d^2 (R_0-a)^{-2}-\sin^2\phi } 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{r}{a}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{R_0}{a}-1 \biggr)
\biggl[ \cos\phi  \pm \sqrt{ \biggl(\frac{d}{a}\biggr)^2 \biggl(\frac{R_0}{a}-1 \biggr)^{-2}-\sin^2\phi } \biggr]
</math>
  </td>
</tr>
</table>

and, the coordinates of points along the surface of the torus <math>~(\varpi,z)</math> are provided by the expressions,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\varpi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
a + (R_0 - a)\cos\phi \biggl[
\cos\phi  \pm \sqrt{ d^2 (R_0 - a)^{-2}-\sin^2\phi }
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(R_0 - a)\sin\phi \biggl[
\cos\phi  \pm \sqrt{ d^2 (R_0 - a)^{-2}-\sin^2\phi }
\biggr]
</math>
  </td>
</tr>
</table>
We have tested this pair of expressions using Excel and have successfully demonstrated that they do, indeed, trace out a circle of radius, <math>~d</math>, whose center is offset from the symmetry axis by a distance, <math>~R_0</math>.

====Set of Circles Whose Offset Increases With Circle Diameter====

A set of nested off-center circles will be described by allowing <math>~R_0 = R_0(d)</math>, that is, by having the off-set distance, <math>~R_0</math>, vary with the size of the circle, <math>~d</math>.  The above prescription for the normalized "coordinate" <math>~r/a</math> will work for ''any'' prescribed <math>~R_0(d)</math> function.

But a ''particular'' <math>~R_0(d)</math> function is demanded if we want this derived prescription to represent the behavior of toroidal coordinates.  In a [[Apps/DysonWongTori#Introducing_Toroidal_Coordinates|toroidal coordinate system]], a specification of the value of the "radial" coordinate, <math>~\eta</math>, automatically dictates the ratio <math>~R_0/d</math>; but we are not at liberty to separately define the value of the ''difference,'' <math>~(R_0 - d)</math>.  Instead, we must enforce the toroidal-coordinate relation,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~R_0^2 - d^2</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ \frac{R_0}{a}-1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ 1 + \delta^2\biggr]^{1 / 2} -1 \, ,</math>
  </td>
</tr>
</table>
where we have adopted the shorthand notation, <math>~\delta\equiv d/a</math>.  Hence,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{r}{a}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~[ \sqrt{1+\delta^2} -1 ]
\{ \cos\phi  \pm [\delta^2 ( \sqrt{1+\delta^2} -1 )^{-2}-\sin^2\phi ]^{1 / 2} \}
</math>
  </td>
</tr>
</table>

Now, in a [[Apps/DysonWongTori#Introducing_Toroidal_Coordinates|toroidal coordinate system]], there is a similar "radial" coordinate, <math>~\eta</math>, whose value varies with distance from the ''anchor ring'' of radius, <math>~a</math>.  Its value depends on both <math>~R_0</math> and <math>~d</math> via the relation,
<div align="center">
<math>~R_0 = d\cosh\eta \, .</math>
</div>
This means that, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\cosh\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\delta}\biggl(\frac{R_0}{a}\biggr) = \frac{\sqrt{1+\delta^2}}{\delta} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ \delta^2 \cosh^2\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 + \delta^2</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ \delta^2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\cosh^2\eta - 1} = \frac{1}{\sinh^2\eta} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ \sqrt{1  + \delta^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[1 + \frac{1}{\sinh^2\eta} \biggr]^{1 / 2}
= \coth\eta
\, ,</math>
  </td>
</tr>
</table>
which also means that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{r}{a}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~[ \coth\eta -1 ]
\biggl\{ \cos\phi  \pm \biggl[ ( \cosh\eta -\sinh\eta )^{-2} -\sin^2\phi \biggr]^{1 / 2}
\biggr\} \, .
</math>
  </td>
</tr>
</table>

====Case of Small Offset====

Another way to look at this issue is to go [[#Dsquared|back to the expression]],
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~d^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
(R_\mathrm{JPO}-\varpi_0)^2 
+ 2\biggl[ (R_\mathrm{JPO}-\varpi_0) r\cos\phi \biggr]
+r^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \delta^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~\biggl(\frac{r}{a}\biggr)^2
+ \frac{r}{a}\biggl[ 2\biggl(1 - \frac{R_0}{a}\biggr)\biggr] \cos\phi 
+ \biggl(1 - \frac{R_0}{a}\biggr)^2 
</math>
  </td>
</tr>
</table>
and assume that, while still dependent on the radial coordinate, the dimensionless offset is small.  That is, assume that,
<div align="center">
<math>~\Delta(\delta) \equiv 1 - \frac{R_0(\delta)}{a} \ll 1 \, .</math>
</div>
In this case, we can write,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ \delta^2</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
 <td align="left">
<math>~\biggl(\frac{r}{a}\biggr)^2
+ 2\Delta(\delta) \biggl( \frac{r}{a} \biggr)  \cos\phi 
+\cancelto{0}{\Delta^2(\delta)} \, .
</math>
  </td>
</tr>
</table>
And differentiating both sides of the expression with respect to <math>~r/a</math> gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0 </math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~2\biggl(\frac{r}{a}\biggr) + 2\Delta(\delta) \cos\phi</math>
  </td>
</tr>
</table>

<font color="red">'''COMMENT by Tohline'''</font> (15 August 2018):  I'm not sure that this is leading where I had hoped.  I am gearing up to draw a comparison between these last expressions and eq. (74) in [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Ostriker's (1964) Paper II].

<!--
===First Attempt===
Based on my (initial, casual) study of this paper, Figure 1 appears to illustrate a configuration in which the density is constant on various nested toroidal surfaces such that, working from the highest density location, outward, the location <math>~R</math> of the center of each torus shifts to larger and larger values.  It would therefore appear as though Ostriker's <math>~R</math> must be a function of the density-marker.  Using the subscript, <math>~i</math>, as the marker, we represent the density as a function, <math>~\rho(r_i)</math>, and recognize that <math>~R = R(r_i)</math> as well.

We recognize that the radial coordinate, <math>~\eta</math>, in a toroidal-coordinate system behaves in this same manner.  Each <math>~\eta = ~ \mathrm{const}</math> surface is a circle of radius, <math>~d</math>, whose center is located a distance from the symmetry axis, <math>~R_0 = \sqrt{a^2 + d^2}</math>.  And, holding <math>~a</math> fixed, the accompanying definition is,
<div align="center">
<math>~\cosh\eta = \frac{R_0}{d} =\biggl[ 1 + \frac{a^2}{d^2} \biggr]^{1 / 2} = \frac{1}{\delta}\biggl[1 + \delta^{2} \biggr]^{1 / 2} \, ,</math>
</div>
where, <math>~\delta \equiv d/a</math>.  Comparing Ostriker's notation with a [[2DStructure/ToroidalGreenFunction#Basic_Elements_of_a_Toroidal_Coordinate_System|toroidal coordinate system]] whose ''anchor ring'' is at the meridional-plane location <math>~(\varpi,z) = (a,0)</math>, we find that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~R+r\cos\phi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a\sinh\eta}{(\cosh\eta - \cos\theta)} \, ,</math>&nbsp; &nbsp; &nbsp; and,
  </td>
</tr>

<tr>
  <td align="right">
<math>~r\sin\phi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a\sin\theta}{(\cosh\eta - \cos\theta)} \, .</math>
  </td>
</tr>
</table>
It appears that we can make the following direct associations: &nbsp; <math>~R_0  \leftrightarrow R_\mathrm{JPO}</math> and <math>~d \leftrightarrow r_\mathrm{JPO}</math>.  Hence, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d\sin\phi}{R_0+d\cos\phi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\sin\theta}{\sinh\eta}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\sin\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{d \sinh\eta \sin\phi}{R_0+d\cos\phi} \, .</math>
  </td>
</tr>
</table>
And,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~R_0+d\cos\phi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a\sinh\eta}{(\cosh\eta - \cos\theta)} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\cos\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\cosh\eta - \frac{a\sinh\eta}{R_0+d\cos\phi} \, .</math>
  </td>
</tr>
</table>
Putting these together we find that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~1 = \sin^2\theta + \cos^2\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{d \sinh\eta \sin\phi}{R_0+d\cos\phi}  \biggr]^2 + \biggl[ \cosh\eta - \frac{a\sinh\eta}{R_0+d\cos\phi} \biggr]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ (R_0 + d\cos\phi)^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
[d \sinh\eta \sin\phi]^2 + [ \cosh\eta(R_0 + d\cos\phi) - a\sinh\eta]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \biggl[ \frac{R_0}{d} + \cos\phi\biggr]^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
[\sinh\eta \sin\phi]^2 + \biggl[ \cosh\eta \biggl(\frac{R_0}{d} + \cos\phi \biggr) - \frac{a}{d} \sinh\eta \biggr]^2
</math>
  </td>
</tr>
</table>
-->