Editing Apps/Ostriker64 (section)

====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>