Editing Appendix/Ramblings/ToroidalCoordinates (section)

====Initial Contact====
Geometrically we appreciate that, as <math>~r_0</math> is increased, the two circles will first touch at a point that lies along the (blue-dashed) line-segment that connects the centers of both circles.  More specifically, the initial interception will be at the point identified in Figure 2 by the solid blue dot lying on the surface of the pink torus.  The distance between the two centers &#8212; which we will denote as <math>~h</math> &#8212; is also the hypotenuse of a right triangle whose other two sides are of length (opposite the angle, <math>~\alpha</math>) <math>~\varpi_t - R_0</math> and (adjacent to the angle, <math>~\alpha</math>) <math>~Z_0</math>.  We see that the initial interception will occur when <math>~r_0 + r_t = h</math>, that is, when 
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<math>~r_0 = r_+</math>
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<math>~\equiv</math>
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<math>~h - r_t</math>
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&nbsp;
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<math>~=</math>
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<math>~[(\varpi_t - R_0)^2 + Z_0^2 ]^{1/2} - r_t </math>
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&nbsp;
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<math>~=</math>
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<math>~(\varpi_t - R_0)[1 + \Lambda^2 ]^{1/2} - r_t \, ,</math>
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where,
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<math>~\Lambda</math>
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<math>~\equiv</math>
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<math>~\frac{Z_0}{\varpi_t - R_0} \, .</math>
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For later reference, we note that the cylindrical coordinates associated with this initial point of contact &#8212; ''i.e.,'' the point identified in Figure 2 by the solid blue dot lying on the surface of the pink torus &#8212; are,
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<math>~\varpi_+</math>
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<math>~=</math>
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<math>~\varpi_t - r_t \sin\alpha</math>
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&nbsp;
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<math>~=</math>
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<math>~\varpi_t - \frac{r_t (\varpi_t-R_0)}{[(\varpi_t - R_0)^2 + Z_0^2 ]^{1/2} }</math>
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&nbsp;
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<math>~=</math>
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<math>~\varpi_t - \frac{r_t }{[1+\Lambda^2 ]^{1/2} } \, ,</math>
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and,
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<math>~Z_+</math>
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<math>~=</math>
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<math>~r_t \cos\alpha</math>
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&nbsp;
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<math>~=</math>
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<math>~\frac{r_t Z_0}{[(\varpi_t - R_0)^2 + Z_0^2 ]^{1/2} }</math>
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&nbsp;
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<math>~=</math>
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<math>~\frac{r_t \Lambda}{[1+\Lambda^2 ]^{1/2} } \, .</math>
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