Editing 2DStructure/ToroidalCoordinateIntegrationLimits (section)

===Special Case Alignment===
It should perhaps not be surprising to find that a toroidal coordinate system can be effectively used to quantitatively describe some properties &#8212; certainly the volume and possibly the gravitational potential &#8212; of circular tori because each <math>\xi_1 = </math> constant surface in a toroidal coordinate system (see the black circles in the right-hand panel of Figure 1) defines the surface of an axisymmetric, circular torus.  In order to map from a cylindrical-coordinate representation of a circular torus (see the left-hand panel of Figure 1) to a toroidal-coordinate representation of that torus, at first glance it would seem reasonable to establish the alignment depicted schematically in Figure 2.  That is, align the horizontal and vertical axes of the bipolar coordinate system (right-hand panel of Figure 1) with, respectively, the <math>\varpi-</math>axis and the <math>Z-</math>axis of the cylindrical coordinate system, then scale the overall size of the bipolar coordinate system until one <math>\xi_1 = </math> constant surface perfectly aligns with the surface of the (pink) torus.  

<table border="1" cellpadding="8" align="center" width="300px">
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  <th align="center" colspan="1"><font size="+1">Figure 2:</font>
Schematic Illustration of "Special Case" Alignment</th>
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[[File:MergedTorus01.png|300px|Merged image of (pink) circular torus and toroidal-coordinate system]]
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Here, a mapping from cylindrical coordinates to toroidal coordinates is achieved by aligning the horizontal and vertical axes of the bipolar coordinate system (right-hand panel of Figure 1) with, respectively, the <math>\varpi-</math>axis and the <math>Z-</math>axis of the cylindrical coordinate system, then scaling the overall size of the bipolar coordinate system, <math>~a</math>, until one <math>\xi_1 = </math> constant surface perfectly aligns with the surface of the (pink) torus.  
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This is the "special case" alignment that we have [[2DStructure/ToroidalCoordinates#Special_Case|discussed in an accompanying chapter]].  It is achieved by setting the scale-length, <math>~a</math>, of the toroidal-coordinate system to a value given by the expression, 
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<math>~a</math>
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<math>~=</math>
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<math>~\sqrt{\varpi_t^2 - r_t^2}\, ,</math>
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which means that the "radial" toroidal coordinate that aligns with the surface of the (pink) torus has the value,
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<math>~\xi_1</math>
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<math>~=</math>
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<math>~\frac{\varpi_t}{r_t} \, .</math>
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While the ''special'' alignment depicted in Figure 2 might at first glance seem reasonable, we have found that it does not generally enable us to transform the multi-dimensional integral expression for the gravitational potential into a one-dimensional integral over <math>~\xi_1</math>, as desired.