Editing Apps/DysonWongTori (section)

====Some Geometric Relations====
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'''Modification of MacMillan's Figure 61'''
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[[File:ModifiedMacMillan61.png|460px|center|Modification of MacMillan's Figure 61]]
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Throughout his derivation, [https://www.amazon.com/Theory-Potential-W-D-Macmillan/dp/0486604861/ref=sr_1_2?s=books&ie=UTF8&qid=1503444466&sr=1-2&keywords=the+theory+of+the+potential MacMillan (1958)] uses the parameter, <math>~a</math>, to represent the radius of the circular "hoop" &#8212; that is, the distance from the center of the hoop to either point <math>~A</math> or point <math>~B</math> as marked in both Figure 60 and Figure 61.  In the diagram presented here, on the right, we have modified his Figure 61 (modifications are in red) to explicitly identify two additional lengths that will come into play when we reference toroidal coordinates, below: &nbsp; The parameter, <math>~R</math>, identifies the distance from the center of the hoop to the center of the meridional-plane circle; and the parameter, <math>~d</math>, identifies the radius of this meridional-plane circle.  (Note that the distance between point <math>~A</math> and the center of the meridional-plane circle is <math>~R-a</math>.)  Given that the meridional-plane circle has been drawn in such a way that the ratio, <math>~\rho_1/\rho_2 = c</math>, at all points <math>~P</math> along the circle, a useful relationship can be derived between the three parameters, <math>~R, d</math> and <math>~a</math> as follows.  

If <math>~P</math> is moved around the circle to align with point <math>~D</math>, we can write,
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<math>~\rho_1 = \rho_2 c = d + (R-a) \, ,</math>
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&nbsp; &nbsp; and, &nbsp; &nbsp; 
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<math>~\rho_2 = 2a + (R-a) + d </math>
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<math>~\Rightarrow ~~~ 
c = \frac{d + (R-a)}{2a + (R-a) + d} \, .
</math>
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Similarly, if <math>~P</math> is moved around the circle to align with point <math>~C</math>, we can write,
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<math>~\rho_1 = \rho_2 c = d - (R-a) \, ,</math>
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&nbsp; &nbsp; and, &nbsp; &nbsp; 
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<math>~\rho_2 = 2a - [ d - (R-a)]</math>
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<math>~\Rightarrow ~~~ 
c = \frac{d - (R-a)}{ 2a - [ d - (R-a)]} \, .
</math>
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Equating these two expressions for <math>~c</math> then gives,
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<math>~\frac{d + R-a}{a + R + d}</math>
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<math>~=</math>
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<math>~\frac{d - R + a}{ a  -d + R}</math>
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<math>~\Rightarrow ~~~
(d + R-a)(a  -d + R )
</math>
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<math>~=</math>
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<math>~
(d - R + a)(a + R + d )
</math>
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<math>~\Rightarrow ~~~
[R - (a- d) ][R + (a  -d) ]
</math>
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<math>~=</math>
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<math>~
[(a + d) - R ][(a + d) + R ]
</math>
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<math>~\Rightarrow ~~~
R^2 - (a- d)^2
</math>
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<math>~=</math>
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<math>~
(a + d)^2 - R^2 
</math>
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<math>~\Rightarrow ~~~
2R^2 
</math>
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<math>~=</math>
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<math>~
(a + d)^2 + (a- d)^2 
</math>
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&nbsp;
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<math>~=</math>
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<math>~
2a^2 + 2d^2 \, ,
</math>
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<span id="TorusGeometry">or, finally,</span>
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<math>~a^2 = R^2 - d^2 \, .</math>
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Similarly, it can be shown that,
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<math>~c</math>
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<math>~=</math>
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<math>~\biggl[1 - \biggl(\frac{R-d}{R+d}\biggr)^{1 / 2}  \biggr]\biggl[1 + \biggl(\frac{R-d}{R+d}\biggr)^{1 / 2}   \biggr]^{- 1 }</math>
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&nbsp;
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<math>~=</math>
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<math>~\frac{R}{d} \biggl[ 1 - \sqrt{1 - \frac{d^2}{R^2}} \biggr] \, .</math>
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Thus, the ''aspect ratio'',
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<math>~\frac{R}{d}</math>
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<math>~=</math>
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<math>~\frac{1+c^2}{2c} = \frac{1}{2}\biggl[ \frac{\rho_2}{\rho_1} + \frac{\rho_1}{\rho_2} \biggr] \, .</math>
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