Revision as of 00:31, 10 November 2015

Toroidal-Coordinate Integration Limits

In support of our accompanying discussion of the gravitational potential of a uniform-density circular torus, here we explain in detail what limits of integration must be specified in order to accurately determine the volume — and, hence also the total mass — of such a torus using toroidal coordinates.

Apollonian Circles (schematic) (see also Wikipedia's Apollonian Circles)

Quantitative Illustration of Employed Toroidal Coordinate System

Schematic Zones
Zone I <math>~Z_0 > r_t</math> for any <math>~a</math>	Zone II <math>~r_t > Z_0 > 0</math> and <math>~a < (\varpi_t-r_t)</math>	Zone II <math>~r_t > Z_0 > 0</math> and <math>~a < (\varpi_t-r_t)</math>

@@ Line 39: / Line 39: @@
 <table border="1" cellpadding="8" align="center">
 <tr>
-   <th align="center" colspan="2"><font size="+1">Schematic Zones</font></th>
+   <th align="center" colspan="3"><font size="+1">Schematic Zones</font></th>
 </tr>
 <tr>
    <th align="center" colspan="1"><font size="+1">Zone I</font><p></p>
 <math>~Z_0 > r_t</math>&nbsp; for any <math>~a</math></th>
+  <th align="center" colspan="1"><font size="+1">Zone II</font><p></p>
+<math>~r_t > Z_0 > 0</math>&nbsp; and &nbsp; <math>~a < (\varpi_t-r_t)</math></th>
    <th align="center" colspan="1"><font size="+1">Zone II</font><p></p>
 <math>~r_t > Z_0 > 0</math>&nbsp; and &nbsp; <math>~a < (\varpi_t-r_t)</math></th>
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 <td align="center">
 [[File:Apollonian_myway7B.png|300px|Apollonian Circles]]
+</td>
+<td align="center">
+[[File:Apollonian_myway8B.png|300px|Apollonian Circles]]
 </td>
 </tr>