Editing 2DStructure/ToroidalCoordinates (section)

===Chosen Test Mass Distribution===

For purposes of illustration, we will follow the lead of [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, Hur&eacute; &amp; Hersant (2012)] &#8212; see the left-hand panel of the following figure ensemble &#8212; and seek to determine the gravitational potential, both inside and outside, of a uniform-density, equatorial-plane torus whose (pink) meridional cross-section is exactly a circle.  More specifically, as illustrated in our Figure 1 &#8212; see the right-hand panel of the following figure ensemble &#8212; at all azimuthal angles, a cross-section through the (pink) torus is prescribed by the familiar algebraic expression for an off-center circle, namely,
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<math>~(\varpi_t - \varpi)^2 + Z^2</math>
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<math>~=</math>
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<math>~r_t^2 \, .</math>
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Everywhere inside this toroidal surface we set <math>~\rho(\varpi, Z) = \rho_0</math>, that is, the density is uniform with the value, <math>~\rho_0</math>.

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Figure 4 extracted without modification from p. 2640 of [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, Hur&eacute; &amp; Hersant (2012)]
<br />
<i>"The Potential of Discs from a 'Mean Green Function'"</i><br />
Monthly Notices of the Royal Astronomical Society, vol. 424, pp. 2635-2645 &copy; RAS
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<th align="center"><font size="+1">Our Figure 1</font></th>
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[[File:Figure4THH2012.png|350px|To be inserted: Fig. 4 from Trova, Hur&eacute; &amp; Hersant (2012)]]
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[[File:DiagramToroidalCoordinates.png|350px|Diagram of Torus and Toroidal Coordinates]]
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<span id="OffCenterCircle">Notice that another off-center circle &#8212; this one with a purple perimeter and otherwise white, rather than pink &#8212; appears in our Figure 1 diagram.  In the discussion that follows, it will be used to represent the meridional-plane cross-section of one axisymmetric surface in an MF53 toroidal-coordinate system.  Here we simply point out that this "surface" is also prescribed by an algebraic expression for an off-center circle, namely,</span>
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  <td align="right">
<math>~(R_0 - \varpi)^2 + (Z_0 - Z)^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r_0^2 \, .</math>
  </td>
</tr>
</table>
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