Editing 2DStructure/ToroidalCoordinateIntegrationLimits (section)

===Practical Transformation===
However, we have discovered that this desired transformation can be accomplished by shifting the toroidal-coordinate system vertically and scaling it such that its off-axis "origin" aligns with the cylindrical-coordinate location, <math>~(R_0, Z_0)</math>, at which the gravitational potential is to be evaluated.  The left-hand panel of Figure 3 illustrates what results from aligning in this manner the left-hand and right-hand panels of Figure 1.  The origin of the toroidal coordinate system sits on the green line-segment, a distance <math>~Z = Z_0</math> above the equatorial plane of the torus, and a distance <math>~a = R_0</math> from the symmetry axis.

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  <th align="center" colspan="3"><font size="+1">Figure 3:  </font>Quantitative Illustration of Employed Toroidal Coordinate System</th>
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[[File:Apollonian_myway5B.png|240px|Apollonian Circles]]
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[[File:TCoordsE.gif|240px|Diagram of Torus and Toroidal Coordinates]]
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[[File:ConstantXi2.png|240px|Diagram of Torus and xi_2-constant Toroidal Coordinate curve]]
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The middle and right-hand panels of Figure 3 display quantitatively how the boundary of the (pink) circular torus is defined in toroidal coordinates when an alignment of coordinate systems is made, as illustrated in the left-hand panel of Figure 3, for the parameter values:  <math>~(R_0, Z_0) = (\tfrac{1}{3}, \tfrac{3}{4})</math> and <math>~(\varpi_t, r_t) = (\tfrac{3}{4}, \tfrac{1}{4})</math>.  As the middle panel shows, black toroidal-coordinate circles intersect and/or thread through the (pink) torus for values of the radial coordinate in the range, <math>1.045 \leq \xi_1 \leq 1.193</math>.  And, as the right-hand panel shows, the red toroidal-coordinate circle that corresponds to the angular-coordinate value, <math>~\xi_2 = 0.885198</math>, not only threads through the (pink) torus but identifies the "angle" at which the two limiting <math>\xi_1 =</math> constant circles touch the surface of the torus.  The derivations that have led to the construction of these two figure panels are presented in [[2DStructure/ToroidalCoordinates#Identifying_Limits_of_Integration|an accompanying discussion]].