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=Toroidal-Coordinate Integration Limits= In support of our [[2DStructure/ToroidalCoordinates#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential| 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. ==Mapping from Cylindrical to Toroidal Coordinates== Referencing the illustration displayed in the left-hand panel of Figure 1, our goal is to determine the gravitational potential at any cylindrical-coordinate location <math>~(R_0, Z_0)</math> due to a uniform-density circular torus whose major radius is <math>~\varpi_t</math> and whose minor, cross-sectional radius is <math>~r_t</math>. Here we explain how a toroidal coordinate system <math>~(\xi_1, \xi_2)</math> — as defined, for example, by [[Appendix/References#MF53|MF53]] and as illustrated schematically in the right-hand panel of Figure 1 — can be used to reduce the geometric complexity of this problem. Note that, in our illustration, each black circle identifies a <math>\xi_1 = </math> constant surface and each red circle identifies a <math>\xi_2 = </math> constant surface. Note, as well, that the overall length scale (not labeled in our schematic diagram) is set by the distance, <math>~a</math>, from the vertical symmetry axis to the origin of the toroidal coordinate system — the point (actually, the axisymmetric ring) at which all the red circles intersect one another. Here we will show how, when using an appropriately aligned toroidal coordinate system, the three-dimensional, weighted integral over the mass distribution that is required to determine the gravitational potential at any location can be reduced to the sum of a small number (1 - 4) of one-dimensional integrals over the "radial" coordinate, <math>~\xi_1</math>. <table border="1" cellpadding="8" align="center"> <tr> <th align="center" colspan="2"><font size="+1">Figure 1: Meridional slice through …</font></th> </tr> <tr> <th align="center" colspan="1"><font size="+1">(Pink) Circular Torus</font></th> <th align="center" colspan="1"><font size="+1">Toroidal Coordinate System (schematic)</font><p></p> (see also [https://en.wikipedia.org/wiki/File:Apollonian_circles.svg Wikipedia's Apollonian Circles])</th> </tr> <tr> <td align="center"> [[File:SimpleTorusIllustration.png|370px|Torus Illustration]] </td> <td align="center"> [[File:Apollonian_myway4.png|370px|Apollonian Circles]] </td> </tr> <tr> <td align="left"> The pink circle represents the meridional cross-section through an axisymmetric, circular torus that lies in the equatorial plane of a cylindrical <math>~(\varpi, Z)</math>, coordinate system. The torus has a major axis of length, <math>~\varpi_t</math>, and a minor, cross-sectional radius of length, <math>~r_t</math>. The red circular dot identifies the cylindrical-coordinate location, <math>~(R_0, Z_0)</math>, at which the gravitational potential is to be evaluated. </td> <td align="left" width="50%"> As is explained more fully in [https://en.wikipedia.org/wiki/Toroidal_coordinates#Coordinate_surfaces Wikipedia's discussion of toroidal coordinates], rotating this two-dimensional bipolar coordinate system about the vertical axis produces a three-dimensional toroidal coordinate system. Black circles centered on the horizontal axis become circular tori — each identifying an axisymmetric, <math>\xi_1 =</math> constant surface; whereas, red circles centered on the vertical axis become spheres — each identifying an axisymmetric, <math>\xi_2 =</math> constant surface. </td> </tr> </table> ===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 — certainly the volume and possibly the gravitational potential — 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"> <tr> <th align="center" colspan="1"><font size="+1">Figure 2:</font> Schematic Illustration of "Special Case" Alignment</th> </tr> <tr> <td align="center"> [[File:MergedTorus01.png|300px|Merged image of (pink) circular torus and toroidal-coordinate system]] </td> </tr> <tr> <td align="left"> 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. </td> </tr> </table> 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, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~a</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\sqrt{\varpi_t^2 - r_t^2}\, ,</math> </td> </tr> </table> </div> which means that the "radial" toroidal coordinate that aligns with the surface of the (pink) torus has the value, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\xi_1</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{\varpi_t}{r_t} \, .</math> </td> </tr> </table> </div> 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. ===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. <table border="1" cellpadding="8" align="center"> <tr> <th align="center" colspan="3"><font size="+1">Figure 3: </font>Quantitative Illustration of Employed Toroidal Coordinate System</th> </tr> <tr> <td align="center"> [[File:Apollonian_myway5B.png|240px|Apollonian Circles]] </td> <td align="center"> [[File:TCoordsE.gif|240px|Diagram of Torus and Toroidal Coordinates]] </td> <td align="center"> [[File:ConstantXi2.png|240px|Diagram of Torus and xi_2-constant Toroidal Coordinate curve]] </td> </tr> </table> 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]]. ==One-Dimensional Integral Equations== ===Volume of Circular Torus=== <div align="center"> <table border="1" cellpadding="8"> <tr><td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\frac{V_i}{V_\mathrm{torus}}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{a^3}{2\pi \varpi_t r_t^2} \int\limits_{\xi_1 = \lambda_i}^{\xi_1 = \Lambda_i} d\xi_1 \biggl\{ \frac{(1-\xi_2^2)^{1/2} [ 4\xi_1^2 - 3\xi_1 \xi_2 - 1]}{(\xi_1^2-1)^2 (\xi_1 - \xi_2)^2} + \biggl[ \frac{(2\xi_1^2 + 1)}{(\xi_1^2-1)^{5/2}}\biggr] \cos^{-1}\biggl[ \frac{(\xi_1\xi_2 - 1 )}{(\xi_1- \xi_2)} \biggr] \biggr\}_{\xi_2 = \gamma_i}^{\xi_2 = \Gamma_i} \, . </math> </td> </tr> </table> </td> </tr> </table> </div> Note that when either integration limit, <math>~\gamma_i</math> or <math>~\Gamma_i</math>, is plus one, the integrand inside the curly braces becomes, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\biggl\{~~\biggr\}_{\xi_2 = +1}</math> </td> <td align="center"> <math>~~\rightarrow~~</math> </td> <td align="left"> <math>~ \biggl\{0 + \biggl[ \frac{(2\xi_1^2 + 1)}{(\xi_1^2-1)^{5/2}}\biggr] \cos^{-1}\biggl[ 1 \biggr] \biggr\} = 0 \, . </math> </td> </tr> </table> Alternatively, when either integration limit, <math>~\gamma_i</math> or <math>~\Gamma_i</math>, is minus one, the integrand inside the curly braces becomes, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\biggl\{~~\biggr\}^{\xi_2 = -1}</math> </td> <td align="center"> <math>~~\rightarrow~~</math> </td> <td align="left"> <math>~ \biggl\{0 + \biggl[ \frac{(2\xi_1^2 + 1)}{(\xi_1^2-1)^{5/2}}\biggr] \cos^{-1}\biggl[- 1 \biggr] \biggr\} = \pi \biggl[ \frac{(2\xi_1^2 + 1)}{(\xi_1^2-1)^{5/2}}\biggr] \, . </math> </td> </tr> </table> (In principle, the arccosine of -1 could be negative π, in which case the leading sign on this last expression should be flipped from positive to negative. But empirical evidence tells us that only the positive sign is relevant to this problem.) ===Gravitational Potential due to a Circular Torus=== <div align="center"> <table border="1" cellpadding="8" align="center"> <tr><td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Phi_i(a,Z_0)</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{2^{5/2} G \rho_0 a^{2}}{3} \int\limits_{\xi_1 = \lambda_i}^{\xi_1 = \Lambda_i} \frac{(\xi_1+1)^{1/2}K(\mu) d\xi_1}{(\xi_1^2 - 1)^2 [ (\xi_1^2 - 1)^{1/2}+\xi_1 ]^{1/2} } \biggr[ \frac{\sin \theta(5\xi_1^2 - 4\xi_1 \cos \theta - 1)}{(\xi_1+1)^{1/2} (\xi_1 - \cos \theta)^{3/2}} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </td> <td align="left"> <math>~ - 4\xi_1 E\biggl( \frac{\pi-\theta}{2} \, , \sqrt{\frac{2}{\xi_1 + 1}} \biggr) + (\xi_1-1) F\biggl( \frac{\pi-\theta}{2} \, , \sqrt{\frac{2}{\xi_1 + 1}} \biggr) \biggr]_{\theta = \cos^{-1}(\gamma_i)}^{\theta = \cos^{-1}(\Gamma_i)} \, . </math> </td> </tr> </table> </td></tr> </table> </div> ===Evaluation of Elliptic Integrals=== We use the ''Numerical Recipes'' algorithms to evaluate elliptic integrals; and we have checked our ''use'' of these algorithms against printed tables of elliptic integrals found in my CRC handbook. Here are the relevant steps used in confirming the proper evaluation of these functions. First, my CRC handbook adopts the following notation (evaluating the incomplete elliptic integral of the ''first'' kind): <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~F(k,\phi) = \int_0^\phi \frac{d\Phi}{\sqrt{1 - k^2 \sin^2\Phi}}</math> </td> <td align="center"> and </td> <td align="left"> <math>~ \theta = \sin^{-1}k \, . </math> </td> </tr> </table> </div> Alternatively, in the ''Numerical Recipes'' algorithm, the two arguments of the incomplete elliptic integral of the first kind are, <math>~\mathrm{ellf}(\phi, k)</math>, where the relevant function expression is, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~F(\phi,k)</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \int_0^\phi \frac{d\theta}{\sqrt{1 - k^2 \sin^2\theta}} \, , </math> </td> </tr> </table> </div> and the argument, <math>~\phi</math>, is expected to be provided in radians, instead of in degrees. Some example values are: <table border="1" align="center" cellpadding="8"> <tr> <th align="center" colspan="13">Incomplete Elliptic Integrals of the First & Second Kind</th> </tr> <tr> <td align="center" colspan="6">CRC's <math>~F(k,\phi)</math> (top); <math>~E(k,\phi)</math> (bottom)</td> <td align="center" colspan="1" rowspan="9" bgcolor="#DDDDDD"> </td> <td align="center" colspan="6">Numerical Recipe's <math>~F(\phi,k)</math> (top); <math>~E(\phi,k)</math> (bottom)</td> </tr> <tr> <td align="right"><math>~\theta ~~ \rightarrow</math></td> <td align="center" rowspan="2">20°</td> <td align="center" rowspan="2">35°</td> <td align="center" rowspan="2">50°</td> <td align="center" rowspan="2">65°</td> <td align="center" rowspan="2">80°</td> <td align="right"><math>~k ~~ \rightarrow</math></td> <td align="center" rowspan="2">0.34202014</td> <td align="center" rowspan="2">0.57357644</td> <td align="center" rowspan="2">0.76604444</td> <td align="center" rowspan="2">0.90630779</td> <td align="center" rowspan="2">0.98480775</td> </tr> <tr> <td align="center"><math>~\phi</math></td> <td align="center"><math>~\phi</math><br />(radians)</td> </tr> <tr> <td align="center">11°</td> <td align="right">0.1921<br />0.1918</td> <td align="right">0.1924<br />0.1916</td> <td align="right">0.1927<br />0.1913</td> <td align="right">0.1930<br />0.1910</td> <td align="right">0.1931<br />0.1908</td> <td align="center">0.19198622</td> <td align="right">0.192123431<br />0.191849180</td> <td align="right">0.192373471<br />0.191600363</td> <td align="right">0.192679935<br />0.191296982</td> <td align="right">0.192961053<br />0.191020217</td> <td align="right">0.193140116<br />0.190844685</td> </tr> <tr> <td align="center">23°</td> <td align="right">0.4027<br />0.4002</td> <td align="right">0.4049<br />0.3980</td> <td align="right">0.4078<br />0.3952</td> <td align="right">0.4105<br />0.3927</td> <td align="right">0.4123<br />0.3911</td> <td align="center">0.40142573</td> <td align="right">0.402656868<br />0.400201278</td> <td align="right">0.404940840<br />0.397964813</td> <td align="right">0.407815120<br />0.395213994</td> <td align="right">0.410528813<br />0.392680594</td> <td align="right">0.412298101<br />0.391061525</td> </tr> <tr> <td align="center">35°</td> <td align="right">0.6151<br />0.6067</td> <td align="right">0.6231<br />0.5991</td> <td align="right">0.6336<br />0.5895</td> <td align="right">0.6441<br />0.5806</td> <td align="right">0.6513<br />0.5748</td> <td align="center">0.61086524</td> <td align="right">0.615064063<br />0.606716517</td> <td align="right">0.623082364<br />0.599066179</td> <td align="right">0.633639464<br />0.589516635</td> <td align="right">0.644149296<br />0.580570509</td> <td align="right">0.651323943<br />0.574768972</td> </tr> <tr> <td align="center">56°</td> <td align="right">0.9930<br />0.9622</td> <td align="right">1.0250<br />0.9335</td> <td align="right">1.0725<br />0.8962</td> <td align="right">1.1285<br />0.8595</td> <td align="right">1.1743<br />0.8344</td> <td align="center">0.97738438</td> <td align="right">0.993020463<br />0.962159483</td> <td align="right">1.024993767<br />0.933452135</td> <td align="right">1.072482572<br />0.896220989</td> <td align="right">1.128479892<br />0.859487843</td> <td align="right">1.174295801<br />0.834363150</td> </tr> <tr> <td align="center">77°</td> <td align="right">1.3788<br />1.3104</td> <td align="right">1.4554<br />1.2457</td> <td align="right">1.5867<br />1.1580</td> <td align="right">1.7909<br />1.0643</td> <td align="right">2.0653<br />0.9917</td> <td align="center">1.34390352</td> <td align="right">1.37884243<br />1.31035009</td> <td align="right">1.45540074<br />1.24566028</td> <td align="right">1.58672349<br />1.15795471</td> <td align="right">1.79094197<br />1.06431530</td> <td align="right">2.06528894<br />0.99170032</td> </tr> <tr> <td align="center"><table border="0" cellpadding="3"><tr><td align="center" rowspan="2">(90°)</td><td align="center"><math>~K(k)</math></td></tr><tr><td align="center"><math>~E(k)</math></td></tr></table></td> <td align="right">1.6200<br />1.5238</td> <td align="right">1.7312<br />1.4323</td> <td align="right">1.9356<br />1.3055</td> <td align="right">2.3088<br />1.1638</td> <td align="right">3.1534<br />1.0401</td> <td align="center"><table border="0" cellpadding="3"><tr><td align="center" rowspan="2">(π/2)</td><td align="center"><math>~K(k)</math></td></tr><tr><td align="center"><math>~E(k)</math></td></tr></table></td> <td align="right">1.62002589<br />1.52379921</td> <td align="right">1.73124518<br />1.43229097</td> <td align="right">1.93558110<br />1.30553909</td> <td align="right">2.30878680<br />1.16382796</td> <td align="right">3.15338525<br />1.04011440</td> </tr> </table> ==Integration Limits== We define the following terms that are functions only of the four principal model parameters, <math>~(a, Z_0, \varpi_t, r_t)</math>, and therefore can be treated as constants while carrying out the pair of nested integrals that determine <math>~q_0</math>. ===Zone I:=== <div id="ZoneI"> <table align="center" border="1" cellpadding="8"> <tr> <th align="center" colspan="2"><font size="+1">Figure 4: </font> Zone I</th> </tr> <tr> <td align="center">Definition</td> <td align="center">Schematic Example</td> </tr> <tr> <td align="center"> <math>~Z_0 > r_t</math><p></p>for any <math>~a</math> </td> <td align="center"> [[File:Apollonian_myway5B.png|300px|Apollonian Circles]] </td> </tr> </table> </div> In an [[2DStructure/ToroidalCoordinates#Identifying_Limits_of_Integration|accompanying discussion]], we have derived the following integration limits; numerical values are given for the specific case, <math>~(a, Z_0, \varpi_t, r_t) = (\tfrac{1}{3}, \tfrac{3}{4}, \tfrac{3}{4}, \tfrac{1}{4})</math>: <div align="center" id="LambdaLimits"> <table border="1" cellpadding="5" align="center"> <tr><td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Lambda_1 = \xi_1|_\mathrm{max} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\biggl[1-\biggl( \frac{a}{\varpi_t-\beta_+} \biggr)^2 \biggr]^{-1/2} \approx 1.1927843</math> </td> </tr> <tr> <td align="right"> <math>~\lambda_1 = \xi_1|_\mathrm{min} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\biggl[1-\biggl( \frac{a}{\varpi_t-\beta_-} \biggr)^2 \biggr]^{-1/2} \approx 1.0449467</math> </td> </tr> </table> </td></tr> </table> </div> where, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\kappa</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~ Z_0^2 + a^2 - (\varpi_t^2 - r_t^2) </math> </td> <td align="center"> <math>\rightarrow</math> </td> <td align="left"> <math>~ \frac{5^2}{2^4\cdot 3^2} \approx 0.17361111 </math> </td> </tr> <tr> <td align="right"> <math>~C</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~1 + \biggl( \frac{2Z_0}{\kappa}\biggr)^2 ( \varpi_t^2 - r_t^2) </math> </td> <td align="center"> <math>\rightarrow</math> </td> <td align="left"> <math>~\frac{17 \cdot 1409}{5^4} \approx 38.3248 </math> </td> </tr> <tr> <td align="right"> <math>~\beta_\pm</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~ - \frac{\kappa}{2} \biggl[ \frac{\varpi_t \mp r_t \sqrt{C}}{(\varpi_t + r_t)(\varpi_t - r_t)} \biggr] </math> </td> <td align="center"> <math>\rightarrow</math> </td> <td align="left"> <math>~ -~\frac{5^2}{2^6\cdot 3}\biggl[ 1\mp \sqrt{ \frac{17\cdot 1409}{3^2\cdot 5^4}} \biggr] </math> </td> </tr> </table> Notice that we have specified these integration limits such that, in going from the lower limit <math>~(\lambda_1)</math> to the upper limit <math>~(\Lambda_1)</math>, the value of <math>~\xi_1</math> monotonically increases. <font color="red"><b>CAUTION:</b></font> This statement is often not true. The quantity, <math>~\kappa</math>, changes signs, depending on whether <math>~(a^2 + Z_0^2) \gtrless (\varpi_t^2 -r_t^2)</math>. When <math>~\kappa</math> changes signs, the two quantities, <math>~\xi_1|_\mathrm{max}</math> and <math>~\xi_1|_\mathrm{min}</math> switch roles; specifically, <math>~\xi_1|_\mathrm{max}</math> becomes less than <math>~\xi_1|_\mathrm{min}</math> (or ''visa versa''). Also, <div align="center" id="Parameters"> <table border="1" cellpadding="5" align="center"> <tr><td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Gamma_1 = \xi_2\biggr|_+</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \xi_1 - \frac{ (\xi_1^2-1)^{1/2}}{(\varpi_i/a)_+} </math> </td> </tr> <tr> <td align="right"> <math>~\gamma_1 = \xi_2\biggr|_-</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \xi_1 - \frac{ (\xi_1^2-1)^{1/2}}{(\varpi_i/a)_-} </math> </td> </tr> </table> </td></tr> </table> </div> where, in addition to the quantities defined above, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\biggl(\frac{\varpi_i}{a}\biggr)_\pm</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~\frac{\kappa}{2a^2}\cdot \frac{\mathrm{B}}{\mathrm{A}} \biggl[1 \pm \sqrt{1-\frac{AC}{B^2}} \biggr] </math> </td> </tr> <tr> <td align="right"> <math>~\mathrm{A}</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~\biggl(\frac{Z_0}{a}\biggr)^2 + \biggl[\frac{\varpi_t}{a} - \frac{\xi_1}{(\xi_1^2-1)^{1/2}}\biggr]^2 </math> </td> </tr> <tr> <td align="right"> <math>~\mathrm{B}</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~\biggl(\frac{2\varpi_t Z_0^2}{a\kappa}\biggr) - \biggl[\frac{\varpi_t}{a} - \frac{\xi_1}{(\xi_1^2-1)^{1/2}}\biggr] </math> </td> </tr> </table> Here, our desire also is to specify the integration limits such that <math>~\xi_2</math> monotonically increases in going from the lower limit <math>~(\gamma_1)</math> to the upper limit <math>~(\Gamma_1)</math>. In order to check to see if this is the case, let's test the limiting values of <math>~\xi_2</math> when we are considering a radial-coordinate value roughly midway between its limits, say, when <math>~\xi_1 = 1.1</math>. For this specific case, we find, <div align="center"> <math>~\frac{\xi_1}{(\xi_1^2 - 1)^{1/2}} = 2.400397 \, ;</math><p></p> <math>~\frac{\varpi_t}{a} - \frac{\xi_1}{(\xi_1^2-1)^{1/2}} = - 0.150397 \, ;</math><p></p> <math>~A = 5.085119 \, ;</math><p></p> <math>~B = 14.73040 \, ;</math><p></p> <math>~\biggl(\frac{\varpi_i}{a}\biggr)_\pm = 2.263098\biggl[1 \pm \sqrt{1-0.898157} \biggr] \, ;</math><p></p> <math>~\Rightarrow ~~~ \biggl(\frac{\varpi_i}{a}\biggr)_+ = 2.985318 \, ;</math><p></p> <math>~\Rightarrow ~~~ \biggl(\frac{\varpi_i}{a}\biggr)_- = 1.540878 \, ;</math><p></p> <math>~\Rightarrow ~~~ \xi_2\biggr|_+ = 0.946496 \, ;</math><p></p> <math>~\Rightarrow ~~~ \xi_2\biggr|_- = 0.802600 \, .</math> </div> Hence, our ordering of the limits appears to be the one desired. ===Zone II:=== <div id="ZoneII"> <table align="center" border="1" cellpadding="8"> <tr> <th align="center" colspan="3"><font size="+1">Figure 5: </font> Zone II</th> </tr> <tr> <td align="center">Definition</td> <td align="center">Schematic Example</td> <td align="center">Quantitative Example: Partial Volumes Identified</td> </tr> <tr> <td align="center"> <math>~r_t > Z_0 > 0</math><p></p>and<p></p><math>~a < \varpi_t - \sqrt{r_t^2 - Z_0^2}</math> </td> <td align="center"> [[File:Apollonian_myway7B.png|240px|Apollonian Circles]] </td> <td align="center"> [[File:Zone_II08.png|240px|Zone II Partial Volumes]] </td> </tr> </table> </div> In an [[2DStructure/ToroidalCoordinates#Green_Cropped-Top_Volume|accompanying discussion]], we have derived the following integration limits; example numerical values of various parameters are provided for the specific case where, <math>~(a, Z_0, \varpi_t, r_t) = (\tfrac{1}{3}, \tfrac{3}{20}, \tfrac{3}{4}, \tfrac{1}{4})</math>. Note that, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\kappa</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~- \frac{1319}{2^4\cdot 3^2 \cdot 5^2} \approx -0.36639 \, ;</math> </td> </tr> <tr> <td align="right"> <math>~C</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~1 + \frac{2^5\cdot 3^6 \cdot 5^2}{(1319)^2} = \frac{2322961}{(1319)^2} \approx 1.33522 \, ;</math> </td> </tr> <tr> <td align="right"> <math>~\beta_+</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{1}{2^6\cdot 3^2\cdot 5^2} \biggl[3\cdot 1319 - \sqrt{2322961} \biggr] \approx 0.16895 \, ;</math> </td> </tr> <tr> <td align="right"> <math>~\beta_-</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{1}{2^6\cdot 3^2\cdot 5^2} \biggl[3\cdot 1319 + \sqrt{2322961} \biggr] \approx 0.38063 \, .</math> </td> </tr> </table> </div> <!-- OLDER EXAMPLE In an [[2DStructure/ToroidalCoordinates#Green_Cropped-Top_Volume|accompanying discussion]], we have derived the following integration limits; example numerical values of various parameters are provided for the specific case where, <math>~(a, Z_0, \varpi_t, r_t) = (\tfrac{1}{3}, \tfrac{1}{5}, \tfrac{3}{4}, \tfrac{1}{4})</math>. Note that, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\kappa</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~- \frac{157}{2\cdot 3^2\cdot 5^2} \approx -0.34889 \, ;</math> </td> </tr> <tr> <td align="right"> <math>~C</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{40849}{(157)^2} \approx 1.65723 \, ;</math> </td> </tr> <tr> <td align="right"> <math>~\beta_+</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{1}{2^3\cdot 3^2\cdot 5^2} \biggl[3\cdot 157 - \sqrt{40849} \biggr] \approx 0.14938 \, ;</math> </td> </tr> <tr> <td align="right"> <math>~\beta_-</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{1}{2^3\cdot 3^2\cdot 5^2} \biggl[3\cdot 157 + \sqrt{40849} \biggr] \approx 0.37395 \, .</math> </td> </tr> </table> </div> OLDER EXAMPLE --> ====Partial Volume #II-1==== This is the green cropped-top sub-volume identified as Partial Volume #1 (PV#1) in the right-hand panel of [[#ZoneII|Figure 5]], and [[2DStructure/ToroidalCoordinates#Green_Cropped-Top_Volume|discussed in detail elsewhere]]. <div align="center"> <table border="1" cellpadding="5" align="center"> <tr><td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Lambda_1 </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{[\varpi_t - \sqrt{r_t^2 - Z_0^2}]^2 + a^2}{[\varpi_t - \sqrt{r_t^2 - Z_0^2}]^2-a^2} ~\rightarrow~\frac{1489}{13\cdot 53} \approx 2.16110</math> </td> </tr> <tr> <td align="right"> <math>~\lambda_1 </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{[\varpi_t + \sqrt{r_t^2 - Z_0^2}]^2 + a^2}{[\varpi_t + \sqrt{r_t^2 - Z_0^2}]^2-a^2} ~\rightarrow~ \frac{41\cdot 89}{7\cdot 11\cdot 37} \approx 1.28080</math> </td> </tr> </table> </td> <td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Gamma_1 = \xi_2\biggr|_+</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \xi_1 - \frac{ (\xi_1^2-1)^{1/2}}{(\varpi_i/a)_+} </math> </td> </tr> <tr> <td align="right"> <math>~\gamma_1 </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~1</math> </td> </tr> </table> </td> <!-- OMIT Diagram <td align="center"> [[File:CropTopB.png|175px|right|Diagram of "Cropped Top" Torus]] </td> --> </tr> </table> </div> ====Partial Volume #II-2==== This is the sub-volume that is painted yellow and identified as Partial Volume #2 (PV#2) in the right-hand panel of [[#ZoneII|Figure 5]]. <div align="center"> <table border="1" cellpadding="5" align="center"> <tr><td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Lambda_2 </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{[\varpi_t + \sqrt{r_t^2 - Z_0^2}]^2 + a^2}{[\varpi_t + \sqrt{r_t^2 - Z_0^2}]^2-a^2} ~\rightarrow~ \frac{41\cdot 89}{7\cdot 11\cdot 37} \approx 1.28080</math> </td> </tr> <tr> <td align="right"> <math>~\lambda_2 </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\xi_1|_\mathrm{max} = \biggl[1-\biggl( \frac{a}{\varpi_t-\beta_+} \biggr)^2 \biggr]^{-1/2} \approx 1.22088</math> </td> </tr> </table> </td> <td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Gamma_2 = \xi_2\biggr|_+</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \xi_1 - \frac{ (\xi_1^2-1)^{1/2}}{(\varpi_i/a)_+} </math> </td> </tr> <tr> <td align="right"> <math>~\gamma_2 = \xi_2\biggr|_-</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \xi_1 - \frac{ (\xi_1^2-1)^{1/2}}{(\varpi_i/a)_-} </math> </td> </tr> </table> </td></tr> </table> </div> ====Partial Volume #II-3==== This is the sub-volume that is painted orange and identified as Partial Volume #3 (PV#3) in the right-hand panel of [[#ZoneII|Figure 5]]. <div align="center"> <table border="1" cellpadding="5" align="center"> <tr><td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Lambda_3 </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{[\varpi_t - \sqrt{r_t^2 - Z_0^2}]^2 + a^2}{[\varpi_t - \sqrt{r_t^2 - Z_0^2}]^2-a^2} ~\rightarrow~\frac{1489}{13\cdot 53} \approx 2.16110</math> </td> </tr> <tr> <td align="right"> <math>~\lambda_3 </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{[\varpi_t + \sqrt{r_t^2 - Z_0^2}]^2 + a^2}{[\varpi_t + \sqrt{r_t^2 - Z_0^2}]^2-a^2} ~\rightarrow~ \frac{41\cdot 89}{7\cdot 11\cdot 37} \approx 1.28080</math> </td> </tr> </table> </td> <td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Gamma_3 </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 </math> </td> </tr> <tr> <td align="right"> <math>~\gamma_3 = \xi_2\biggr|_-</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \xi_1 - \frac{ (\xi_1^2-1)^{1/2}}{(\varpi_i/a)_-} </math> </td> </tr> </table> </td></tr> </table> </div> ====Partial Volume #II-4==== This is the sub-volume that is painted blue and identified as Partial Volume #4 (PV#4) in the right-hand panel of [[#ZoneII|Figure 5]]. <div align="center"> <table border="1" cellpadding="5" align="center"> <tr><td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Lambda_4 </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\xi_1|_\mathrm{min} = \biggl[1-\biggl( \frac{a}{\varpi_t-\beta_-} \biggr)^2 \biggr]^{-1/2} \approx 2.32125</math> </td> </tr> <tr> <td align="right"> <math>~\lambda_4 </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{[\varpi_t - \sqrt{r_t^2 - Z_0^2}]^2 + a^2}{[\varpi_t - \sqrt{r_t^2 - Z_0^2}]^2-a^2} ~\rightarrow~\frac{1489}{13\cdot 53} \approx 2.16110</math> </td> </tr> </table> </td> <td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Gamma_4 = \xi_2\biggr|_+</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \xi_1 - \frac{ (\xi_1^2-1)^{1/2}}{(\varpi_i/a)_+} </math> </td> </tr> <tr> <td align="right"> <math>~\gamma_4 = \xi_2\biggr|_-</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \xi_1 - \frac{ (\xi_1^2-1)^{1/2}}{(\varpi_i/a)_-} </math> </td> </tr> </table> </td></tr> </table> </div> ====Summary (Zone II)==== In summary, for a given set of the three model parameters <math>~(a, \varpi_t, r_t)</math> and the fourth, <math>~Z_0</math>, in the range, <math>~0 < Z_0 < r_t</math>, the volume of the (pink) circular torus is determined by adding together four partial volumes — that is, adding together the results of four separate 1D integrations over the "radial" toroidal coordinate <math>~(\xi_1)</math>. Although a total of eight radial integration limits (four lower limits and four upper limits) are required to fully determine the Zone II torus volume, only four unique limiting values need to be calculated because the partial volumes share <math>~\xi_1</math> boundaries. This has been illustrated by the black vertical dashed and dot-dashed lines in the left-hand panel of Figure 6 — and, drawing from the above discussion, the numerical values of these limits have been recorded in Table 1 — for the ''specific'' case of <math>~Z_0 = \tfrac{3}{20}</math>. <table border="1" cellpadding="8" align="center"> <tr><th align="center" colspan="9"> <font size="+1">Table 1:</font> Zone II Partial Volumes & Integration Limits on <math>~\xi_1</math><p></p> for model parameters <math>~(a, Z_0, \varpi_t, r_t) = (\tfrac{1}{3}, \tfrac{3}{20}, \tfrac{3}{4}, \tfrac{1}{4})</math> </th></tr> <tr> <td align="center" colspan="1"> </td> <td align="center" colspan="2">PV #1</td> <td align="center" colspan="2">PV #2</td> <td align="center" colspan="2">PV #3</td> <td align="center" colspan="2">PV #4</td> </tr> <tr> <td align="center" rowspan="2" colspan="1">Integration<p></p>Limits</td> <td align="center"><math>~\lambda_1</math></td> <td align="center"><math>~\Lambda_1</math></td> <td align="center"><math>~\lambda_2</math></td> <td align="center"><math>~\Lambda_2</math></td> <td align="center"><math>~\lambda_3</math></td> <td align="center"><math>~\Lambda_3</math></td> <td align="center"><math>~\lambda_4</math></td> <td align="center"><math>~\Lambda_4</math></td> </tr> <tr> <td align="center"><math>2.16110</math></td> <td align="center"><math>1.28080</math></td> <td align="center"><math>1.28080</math></td> <td align="center"><math>1.22088</math></td> <td align="center"><math>2.16110</math></td> <td align="center"><math>1.28080</math></td> <td align="center"><math>2.32125</math></td> <td align="center"><math>2.16110</math></td> </tr> <tr> <td align="center" rowspan="1" colspan="1">Volume<p></p>Fraction</td> <td align="center" colspan="2">0.14237851</td> <td align="center" colspan="2">0.21569718</td> <td align="center" colspan="2">0.63537958</td> <td align="center" colspan="2">0.0065448719</td> </tr> <tr> <td align="center" rowspan="1" colspan="1">Total Volume<p></p><font size="-1">(nzones = 5000)</font></td> <td align="center" colspan="8">1.00000014 <math>~\Rightarrow~</math> Error = -1.4E-7 </td> </tr> </table> While the [[#Volume_of_Circular_Torus|radial integrand expression for each partial volume]] is formally the same, it requires a specification of both limits, <math>~\gamma_i</math> and <math>~\Gamma_i</math>, for the "angular" coordinate integration which, as has also just been detailed, vary from one partial volume to the next and generally depend on the value of the radial coordinate. This dependence of the angular coordinate integration limits on the specific value of the radial coordinate across the four separate partial volumes is quantitatively illustrated in Figure 6 for the specific set of model parameters, <math>~(a, \varpi_t, r_t) = (\tfrac{1}{3}, \tfrac{3}{4}, \tfrac{1}{4})</math>, and for twenty-four different values of <math>~Z_0</math>. <table border="1" cellpadding="8" align="center"> <tr><th align="center" colspan="2"> <font size="+1">Figure 6:</font> Zone II Integration Limits on <math>~\xi_2</math><p></p> for model parameters <math>~(a, \varpi_t, r_t) = (\tfrac{1}{3}, \tfrac{3}{4}, \tfrac{1}{4})</math> and various <math>~Z_0</math> </th></tr> <tr> <td align="center"> [[File:Torus_z15B.png|347px|Diagram of Torus and Toroidal Coordinates]] </td> <td align="center"> [[File:Xi2Movie04.gif|Animation of Torus and Toroidal Coordinates]] </td> </tr> <tr> <td align="center" colspan="1" align="center"> <table border="1" align="center" cellpadding="5"> <tr> <td align="center" colspan="4">Example <math>~\xi_2</math> Limits when <math>~Z_0 = \tfrac{3}{20} = 0.15</math></td> </tr> <tr> <td align="center" width="25%"> Partial<p></p>Volume (''i'') </td> <td align="center" width="25%"> Example<p></p> <math>~\xi_1</math> </td> <td align="center" width="25%"> <math>~\gamma_i</math> </td> <td align="center"> <math>~\Gamma_i</math> </td> </tr> <tr> <td align="center" bgcolor="green"><b>#1</b></td> <td align="center">2.00</td> <td align="center">1.00000</td> <td align="center">0.99625</td> </tr> <tr> <td align="center" bgcolor="yellow"><b>#2</b></td> <td align="center">1.25</td> <td align="center">0.91635</td> <td align="center">0.99789</td> </tr> <tr> <td align="center" bgcolor="orange"><b>#3</b></td> <td align="center">2.00</td> <td align="center">0.84538</td> <td align="center">1.00000</td> </tr> <tr> <td align="center" bgcolor="blue"><b>#4</b></td> <td align="center">2.30</td> <td align="center">0.94684</td> <td align="center">0.98886</td> </tr> </table> </td> <td align="center"> </td> </tr> </table> In both panels of Figure 6, at each of a variety of values of the "radial" coordinate, <math>~\xi_1</math> (horizontal axis), a pair of colored dots identify the values (vertical axis) of the two "angular" coordinate integration limits, <math>~\gamma_i</math> (lower value) and <math>~\Gamma_i</math> (upper value). The left-hand panel has been constructed for the specific case, <math>~Z_0 = \tfrac{3}{20} = 0.15</math>; via an animation sequence, the right-hand panel shows how the limits vary with <math>~\xi_1</math> for twenty-four different values of <math>~Z_0</math> in the Zone II range <math>~(0 \leq Z_0 \leq r_t)</math>, as indicated in the lower right-hand corner of each frame. The limits identified by yellow dots must be fed into the radial integrand expression when evaluating partial volume #2 and the blue dots provide the limits for partial volume #4. Green dots identify the lower angular-coordinate integration limit across partial volume #1; orange dots identify the lower limit across partial volume #3; and the upper limit for both partial volume #1 and partial volume #3 is unity (black dots). Four vertical black lines (two dashed and two dot-dashed) have been added to the plot displayed in the left-hand panel of Figure 6 in order to emphasize that the boundaries between the four partial volumes are defined by the radial-coordinate integration limits, <math>~\lambda_i</math> and <math>~\Lambda_i</math>; as has been detailed in Table 1 for the ''specific'' case <math>~(a, Z_0, \varpi_t, r_t) = (\tfrac{1}{3}, \tfrac{3}{20}, \tfrac{3}{4}, \tfrac{1}{4})</math>, the boundaries occur at <math>~\xi_1 = 1.22088, 1.28080, 2.16110,</math> and <math>~2.32125</math>. <div align="center"> <table border="1" cellpadding="5" align="center"> <tr> <th align="center" colspan="7"><font size="+1">Table 1b:</font> Validate Pattern II</th> </tr> <tr><th align="center" colspan="6"> Integration Limits for model parameters <p></p><math>~(a, Z_0, \varpi_t, r_t) = (\tfrac{4}{5}, \tfrac{3}{20}, \tfrac{3}{4}, \tfrac{1}{4})</math> </th> <td align="center" rowspan="7">[[File:Zone_II08.png|250px|Zone II Partial Volumes]]</td> </tr> <tr> <td align="center" colspan="1" rowspan="2"> </td> <td align="center"><math>~\xi_1|_\mathrm{max}</math></td> <td align="center"><math>~\xi_1|_\mathrm{min}</math></td> <td align="center"><math>~\xi_1|_+</math></td> <td align="center"><math>~\xi_1|_-</math></td> <td align="center" rowspan="2"><math>~\infty</math></td> </tr> <tr> <td align="center">1.22088</td> <td align="center">2.32125</td> <td align="center">1.28080</td> <td align="center">2.16110</td> </tr> <!-- TEMPORARY ... <tr> <td align="center">Temporary</td> <td align="center"><b><font color="#FFDD00">END</font></b></td> <td align="center"><b><font color="blue">START</font></b></td> <td align="center"><b><font color="#FFDD00">START</font></b><p></p><b><font color="lightgreen">END</font></b><p></p><b><font color="orange">END</font></b></td> <td align="center"> <b><font color="blue">END</font></b><p></p><b><font color="lightgreen">START</font></b><p></p><b><font color="ORANGE">START</font></b> </td> <td align="center"> </td> </tr> END TEMPORARY --> <tr> <td align="center"><math>~\xi_2|_+</math></td> <td align="center"><b><font color="yellow">END</font></b></td> <td align="center"> </td> <td align="center"> </td> <td align="center"><b><font color="lightgreen">START</font></b><p></p><b><font color="blue">END</font></b></td> <td align="center"> </td> </tr> <tr> <td align="center"><math>~\xi_2|_-</math></td> <td align="center"> </td> <td align="center"><b><font color="blue">START</font></b></td> <td align="center"><b><font color="yellow">START</font></b></td> <td align="center"><b><font color="ORANGE">START</font></b></td> <td align="center"> </td> </tr> <tr> <td align="center"><math>~+1</math></td> <td align="center"> </td> <td align="center"> </td> <td align="center"><b><font color="lightgreen">END</font></b><p></p><b><font color="orange">END</font></b></td> <td align="center"> </td> <td align="center"> </td> </tr> <tr> <td align="center"><math>~-1</math></td> <td align="center"> </td> <td align="center"> </td> <td align="center"> </td> <td align="center"> </td> <td align="center"> </td> </tr> </table> </div> ====Special Case of Z<sub>0</sub> = 0==== As has been recorded in an [[Appendix/Ramblings/ToroidalCoordinates#Tohline.27s_Ramblings|accompanying "Ramblings" appendix]], my inversion of the toroidal-coordinate definitions — which, by default, incorporate the assumption that <math>~Z_0 = 0</math> — has led to the following expressions: <table align="center" border="0" cellpadding="5"> <tr> <td align="right"> <math> ~\xi_1 </math> </td> <td align="center"> <math> ~= </math> </td> <td align="left"> <math> \frac{r(r^2 + 1)} {[\chi^2(r^2-1)^2 + \zeta^2(r^2+1)^2]^{1/2}} \, , </math> </td> </tr> <tr> <td align="right"> <math> ~\xi_2 </math> </td> <td align="center"> <math> ~= </math> </td> <td align="left"> <math> \frac{r(r^2 - 1)}{[\chi^2(r^2-1)^2 + \zeta^2(r^2+1)^2]^{1/2}} \, , </math> </td> </tr> </table> where, <div align="center"> <math> \chi \equiv \frac{\varpi}{a} ~~~;~~~\zeta\equiv\frac{z}{a} ~~~\mathrm{and} ~~~ r=(\chi^2 + \zeta^2)^{1/2} . </math> </div> It seems clear that, in this special case, the first radial coordinate that encounters the ''pink'' torus <math>~(\xi_1|_\mathrm{max})</math> should be given by the cylindrical-coordinate values, <math>~[\varpi,z] = [(\varpi_t - r_t),0]</math>; likewise the ''last'' radial coordinate <math>~(\xi_1|_\mathrm{min})</math> should be given by the cylindrical-coordinate values, <math>~[\varpi,z] = [(\varpi_t + r_t),0]</math>. Hence, in both limits, <math>~\zeta = 0 ~~ \Rightarrow ~~ r = \chi</math>, which further implies, <table align="center" border="0" cellpadding="5"> <tr> <td align="right"> <math> ~\xi_1|_\mathrm{limits} </math> </td> <td align="center"> <math> ~= </math> </td> <td align="left"> <math> \frac{\chi(\chi^2 + 1)} {[\chi^2(\chi^2-1)^2 ]^{1/2}} = \frac{\chi^2 + 1} {\chi^2-1} \, . </math> </td> </tr> </table> Separately, then, we expect, <table align="center" border="0" cellpadding="5"> <tr> <td align="right"> <math> ~\xi_1|_\mathrm{min} </math> </td> <td align="center"> <math> ~= </math> </td> <td align="left"> <math> \frac{(\varpi_t - r_t)^2 + a^2} {(\varpi_t - r_t)^2-a^2} \, , </math> </td> </tr> <tr> <td align="right"> <math> ~\xi_1|_\mathrm{max} </math> </td> <td align="center"> <math> ~= </math> </td> <td align="left"> <math> \frac{(\varpi_t + r_t)^2 + a^2} {(\varpi_t + r_t)^2-a^2} \, . </math> </td> </tr> </table> By comparison, the [[#LambdaLimits|expressions that we have provided, above, for these two limits]] gives, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\biggl[ \xi_1|_\mathrm{max}\biggr]^{-2} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1-\biggl( \frac{a}{\varpi_t-\beta_+} \biggr)^2 \, , </math> </td> </tr> <tr> <td align="right"> <math>~\biggl[ \xi_1|_\mathrm{min} \biggr]^{-2} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1-\biggl( \frac{a}{\varpi_t-\beta_-} \biggr)^2 \, , </math> </td> </tr> </table> where, in the special case of <math>~Z_0 = 0</math>, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\kappa</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ a^2 - (\varpi_t^2 - r_t^2) \, , </math> </tr> <tr> <td align="right"> <math>~\beta_\pm</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ - \frac{\kappa}{2} \biggl[ \frac{\varpi_t \mp r_t }{(\varpi_t + r_t)(\varpi_t - r_t)} \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{1}{2} \biggl[ (\varpi_t^2 - r_t^2) - a^2 \biggr] \biggl[ \frac{1 }{\varpi_t \pm r_t} \biggr] \, . </math> </td> </tr> </table> Hence, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\varpi_t - \beta_\pm</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \varpi_t - \frac{1}{2} \biggl[ (\varpi_t^2 - r_t^2) - a^2 \biggr] \biggl[ \frac{1 }{\varpi_t \pm r_t} \biggr] </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{1}{2(\varpi_t \pm r_t )} \biggl\{ 2\varpi_t(\varpi_t \pm r_t) - \biggl[ (\varpi_t^2 - r_t^2) - a^2 \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{1}{2(\varpi_t \pm r_t )} \biggl\{ 2\varpi_t(\varpi_t \pm r_t) + \biggl[a^2 - (\varpi_t + r_t)(\varpi_t - r_t) \biggr] \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{1}{2} \biggl\{\frac{a^2}{(\varpi_t \pm r_t )} + 2\varpi_t - (\varpi_t \mp r_t) \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{a}{2} \biggl[\frac{a}{(\varpi_t \pm r_t )} + \frac{(\varpi_t \pm r_t)}{a} \biggr] </math> </td> </tr> <tr> <td align="right"> <math>~\Rightarrow ~~~ \biggl( \frac{\varpi_t - \beta_\pm}{a}\biggr)</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl(\frac{\Upsilon_\pm^2 + a^2}{2a\Upsilon_\pm} \biggr) \, , </math> </td> </tr> </table> where, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Upsilon_\pm</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~\varpi_t \pm r_t \, .</math> </td> </tr> </table> </div> This also means that, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~1 - \biggl( \frac{a}{\varpi_t - \beta_\pm}\biggr)^2</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 - \biggl(\frac{2a\Upsilon_\pm}{\Upsilon_\pm^2 + a^2} \biggr)^2 </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{(\Upsilon_\pm^2 + a^2)^2 - 4a^2\Upsilon_\pm^2}{(\Upsilon_\pm^2 + a^2)^2} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl[ \frac{\Upsilon_\pm^2 - a^2}{\Upsilon_\pm^2 + a^2} \biggr]^2 \, . </math> </td> </tr> </table> </div> That is, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\biggl[ \xi_1|_\mathrm{max}\biggr]^{-2} = \biggl[ \frac{\Upsilon_+^2 - a^2}{\Upsilon_+^2 + a^2} \biggr]^2</math> </td> <td align="center"> <math>~\Rightarrow</math> </td> <td align="left"> <math>~\xi_1|_\mathrm{max} = \frac{\Upsilon_+^2 + a^2}{\Upsilon_+^2 - a^2} \, , </math> </td> </tr> <tr> <td align="right"> <math>~\biggl[ \xi_1|_\mathrm{min} \biggr]^{-2} = \biggl[ \frac{\Upsilon_-^2 - a^2}{\Upsilon_-^2 + a^2} \biggr]^2 </math> </td> <td align="center"> <math>~\Rightarrow</math> </td> <td align="left"> <math>~\xi_1|_\mathrm{min} = \frac{\Upsilon_-^2 + a^2}{\Upsilon_-^2 - a^2} \, . </math> </td> </tr> </table> Excellent! This matches the earlier supposition. Also, given that <math>~C = 1</math> and <math>~A = B^2</math>, the pair of values, <math>~(\varpi_i/a)_\pm</math>, is degenerate and given by the expression, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\biggl(\frac{\varpi_i}{a}\biggr)_\pm</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{\kappa}{2a^2 \mathrm{B}} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{1}{2a^2} \biggl[ (\varpi_t^2 - r_t^2) -a^2\biggr] \biggl[\frac{\varpi_t}{a} - \frac{\xi_1}{(\xi_1^2-1)^{1/2}}\biggr]^{-1} \, . </math> </td> </tr> </table> Hence, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\xi_2\biggr|_+</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \xi_1 - \frac{ (\xi_1^2-1)^{1/2}}{(\varpi_i/a)_+} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \xi_1\biggl\{1 - \frac{ (\xi_1^2-1)^{1 / 2}}{\xi_1} \biggl[\frac{\varpi_t}{a} - \frac{\xi_1}{(\xi_1^2-1)^{1/2}}\biggr] 2a^2 \biggl[ (\varpi_t^2 - r_t^2) -a^2\biggr]^{-1} \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \xi_1\biggl\{1 - \biggl[\biggl( \frac{\varpi_t}{a} \biggr)\frac{ (\xi_1^2-1)^{1 / 2}}{\xi_1} - 1 \biggr] 2a^2 \biggl[ (\varpi_t^2 - r_t^2) -a^2\biggr]^{-1} \biggr\} </math> </td> </tr> </table> ===Zone III=== <div align="center" id="Figure7"> <table align="center" border="1" cellpadding="8"> <tr> <th align="center" colspan="3"><font size="+1">Figure 7: </font> Zone III</th> </tr> <tr> <td align="center">Definition</td> <td align="center">Schematic Example</td> <td align="center">Quantitative Example: Partial Volumes Identified</td> </tr> <tr> <td align="center"> <math>~r_t > Z_0 > 0</math><p></p>and<p></p> <math>~\varpi_t - \sqrt{r_t^2 - Z_0^2} < a < \varpi_t + \sqrt{r_t^2 - Z_0^2}</math><p></p> <math>~\biggl[\frac{11}{20} < a < \frac{19}{20}\biggr]</math> </td> <td align="center"> [[File:Apollonian_myway8B.png|240px|Apollonian Circles]] </td> <td align="center"> [[File:LimitsOnTorus2ColoredSmall01.png|240px|Zone II Partial Volumes]] </td> </tr> </table> </div> Here, numerical values will be given for the specific case, <math>~(a, Z_0, \varpi_t, r_t) = (\tfrac{4}{5}, \tfrac{3}{20}, \tfrac{3}{4}, \tfrac{1}{4})</math>: <div align="center"> <table border="1" cellpadding="5" align="center"> <tr><td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\xi_1|_\mathrm{max} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\biggl[1-\biggl( \frac{a}{\varpi_t-\beta_+} \biggr)^2 \biggr]^{-1/2} \approx 8.9243615</math> </td> </tr> <tr> <td align="right"> <math>~\xi_1|_\mathrm{min} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\biggl[1-\biggl( \frac{a}{\varpi_t-\beta_-} \biggr)^2 \biggr]^{-1/2} \approx 1.9116174</math> </td> </tr> </table> </td></tr> </table> </div> where, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\kappa</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~ Z_0^2 + a^2 - (\varpi_t^2 - r_t^2) </math> </td> <td align="center"> <math>\rightarrow</math> </td> <td align="left"> <math>~\frac{13}{2^4\cdot 5} = 0.1625 </math> </td> </tr> <tr> <td align="right"> <math>~C</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~1 + \biggl( \frac{2Z_0}{\kappa}\biggr)^2 ( \varpi_t^2 - r_t^2) </math> </td> <td align="center"> <math>\rightarrow</math> </td> <td align="left"> <math>\frac{457}{13^2} \approx 2.7041420 </math> </td> </tr> <tr> <td align="right"> <math>~\beta_\pm</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~ - \frac{\kappa}{2} \biggl[ \frac{\varpi_t \mp r_t \sqrt{C}}{(\varpi_t + r_t)(\varpi_t - r_t)} \biggr] </math> </td> <td align="center"> <math>\rightarrow</math> </td> <td align="left"> <math>~ -~\frac{1}{2^6\cdot 5 }\biggl[ 39 \mp \sqrt{457} \biggr] </math> </td> </tr> <tr> <td align="right"> <math>~\Rightarrow~~~~\beta_+</math> </td> <td align="center"> <math>\approx</math> </td> <td align="left"> <math>~ -~0.055070130 </math> </td> <td align="right"> </td> <td align="center"> </td> </tr> <tr> <td align="right"> <math>~\beta_-</math> </td> <td align="center"> <math>\approx</math> </td> <td align="left"> <math>~ -~0.188679870 </math> </td> <td align="right"> </td> <td align="center"> </td> </tr> </table> Notice that we have specified these integration limits such that, in going from the lower limit <math>~(\lambda_1)</math> to the upper limit <math>~(\Lambda_1)</math>, the value of <math>~\xi_1</math> monotonically increases. ====Partial Volume #III-1==== This is the sub-volume that is painted blue and identified as Partial Volume #1 (PV#1) in the right-hand panel of [[#ZoneIII|Figure 7]]. <div align="center"> <table border="1" cellpadding="5" align="center"> <tr><td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Lambda_1 </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{a^2 + [\varpi_t - \sqrt{r_t^2 - Z_0^2}]^2}{a^2 - [\varpi_t - \sqrt{r_t^2 - Z_0^2}]^2} ~\rightarrow~ \frac{2^8+11^2}{2^8-11^2} \approx 2.79259</math> </td> </tr> <tr> <td align="right"> <math>~\lambda_1 </math> </td> <td align="center"> <math>~= </math> </td> <td align="left"> <math>~\xi_1|_\mathrm{min} = \biggl[1-\biggl( \frac{a}{\varpi_t-\beta_-} \biggr)^2 \biggr]^{-1/2} \approx 1.9116174</math> </td> </tr> </table> </td> <td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Gamma_1 = \xi_2\biggr|_+</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \xi_1 - \frac{ (\xi_1^2-1)^{1/2}}{(\varpi_i/a)_+} </math> </td> </tr> <tr> <td align="right"> <math>~\gamma_1 = \xi_2\biggr|_- </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \xi_1 - \frac{ (\xi_1^2-1)^{1/2}}{(\varpi_i/a)_-} </math> </td> </tr> </table> </td></tr> </table> </div> ====Partial Volume #III-2==== This is the sub-volume that is painted green and identified as Partial Volume #2 (PV#2) in the right-hand panel of [[#ZoneIII|Figure 7]]. <div align="center"> <table border="1" cellpadding="5" align="center"> <tr><td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Lambda_2 </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{[\varpi_t + \sqrt{r_t^2 - Z_0^2}]^2+a^2}{[\varpi_t + \sqrt{r_t^2 - Z_0^2}]^2-a^2} ~\rightarrow~ \frac{19^2 + 2^8}{19^2 - 2^8} \approx 5.87619</math> </td> </tr> <tr> <td align="right"> <math>~\lambda_2 </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{a^2 + [\varpi_t - \sqrt{r_t^2 - Z_0^2}]^2}{a^2 - [\varpi_t - \sqrt{r_t^2 - Z_0^2}]^2} ~\rightarrow~ \frac{2^8+11^2}{2^8-11^2} \approx 2.79259</math> </td> </tr> </table> </td> <td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Gamma_2 = \xi_2\biggr|_+</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \xi_1 - \frac{ (\xi_1^2-1)^{1/2}}{(\varpi_i/a)_+} </math> </td> </tr> <tr> <td align="right"> <math>~\gamma_2 </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ -1 </math> </td> </tr> </table> </td></tr> </table> </div> ====Partial Volume #III-3==== This is the sub-volume that is painted orange and identified as Partial Volume #3 (PV#3) in the right-hand panel of [[#ZoneIII|Figure 7]]. <div align="center"> <table border="1" cellpadding="5" align="center"> <tr><td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Lambda_3 </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\infty</math> </td> </tr> <tr> <td align="right"> <math>~\lambda_3 </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{[\varpi_t + \sqrt{r_t^2 - Z_0^2}]^2+a^2}{[\varpi_t + \sqrt{r_t^2 - Z_0^2}]^2-a^2} ~\rightarrow~ \frac{19^2 + 2^8}{19^2 - 2^8} \approx 5.87619</math> </td> </tr> </table> </td> <td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Gamma_3 </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ +1 </math> </td> </tr> <tr> <td align="right"> <math>~\gamma_3 </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ -1 </math> </td> </tr> </table> </td></tr> </table> </div> '''Alternative determination of this partial volume:''' Notice that this "orange" meridional-plane segment is a (semi)circle whose cross-sectional radius is (see [[Appendix/Ramblings/ToroidalCoordinates#Walk_Through_Step-By-Step|accompanying discussion]]), <div align="center"> <math>~r_\mathrm{orange} = \frac{a}{\sqrt{\lambda_3^2 - 1}} \, ,</math> </div> and it is associated with a circular torus whose major radius is, <div align="center"> <math>~R_\mathrm{orange} = \lambda_3 r_\mathrm{orange} \, .</math> </div> Hence, using the [[2DStructure/ToroidalCoordinates#Total_Mass|familiar expression for the volume of a circular torus]], we know that the volume associated with this "orange" partial volume is, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~V_\mathrm{orange}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{1}{2}\biggl[ 2\pi^2 R_\mathrm{orange} r_\mathrm{orange}^2 \biggr] = \pi^2 \lambda_3 r_\mathrm{orange}^3 = \frac{\pi^2 a^3 \lambda_3}{(\lambda_3^2-1)^{3/2}} </math> </td> </tr> <tr> <td align="right"> <math>~\Rightarrow~~~~\frac{V_\mathrm{orange}}{V_\mathrm{torus}}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\biggl(\frac{a^3}{2\varpi_t r_t^2}\biggr) \frac{\lambda_3}{(\lambda_3^2-1)^{3/2}} \approx 0.165291952 \, .</math> </td> </tr> </table> </div> ====Partial Volume #III-4==== This is the sub-volume that is painted pink and identified as Partial Volume #4a (PV#4a) in the right-hand panel of [[#ZoneIII|Figure 7]]. <div align="center"> <table border="1" cellpadding="5" align="center"> <tr><td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Lambda_4 </math> </td> <td align="center"> <math>~= </math> </td> <td align="left"> <math>~\xi_1|_\mathrm{max} = \biggl[1-\biggl( \frac{a}{\varpi_t-\beta_+} \biggr)^2 \biggr]^{-1/2} \approx 8.9243615</math> </td> </tr> <tr> <td align="right"> <math>~\lambda_4 </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{a^2 + [\varpi_t - \sqrt{r_t^2 - Z_0^2}]^2}{a^2 - [\varpi_t - \sqrt{r_t^2 - Z_0^2}]^2} ~\rightarrow~ \frac{2^8+11^2}{2^8-11^2} \approx 2.79259</math> </td> </tr> </table> </td> <td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Gamma_{4a} = \xi_2\biggr|_-</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \xi_1 - \frac{ (\xi_1^2-1)^{1/2}}{(\varpi_i/a)_-} </math> </td> </tr> <tr> <td align="right"> <math>~\gamma_{4a} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ -1 </math> </td> </tr> </table> </td></tr> </table> </div> This is the sub-volume that is painted red and identified as Partial Volume #4b (PV#4b) in the right-hand panel of [[#ZoneIII|Figure 7]]. <div align="center"> <table border="1" cellpadding="5" align="center"> <tr><td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Lambda_4 </math> </td> <td align="center"> <math>~= </math> </td> <td align="left"> <math>~\xi_1|_\mathrm{max} = \biggl[1-\biggl( \frac{a}{\varpi_t-\beta_+} \biggr)^2 \biggr]^{-1/2} \approx 8.9243615</math> </td> </tr> <tr> <td align="right"> <math>~\lambda_4 </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{[\varpi_t + \sqrt{r_t^2 - Z_0^2}]^2+a^2}{[\varpi_t + \sqrt{r_t^2 - Z_0^2}]^2-a^2} ~\rightarrow~ \frac{19^2 + 2^8}{19^2 - 2^8} \approx 5.87619</math> </td> </tr> </table> </td> <td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Gamma_{4b} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ +1 </math> </td> </tr> <tr> <td align="right"> <math>~\gamma_{4b} = \xi_2\biggr|_+</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \xi_1 - \frac{ (\xi_1^2-1)^{1/2}}{(\varpi_i/a)_+} </math> </td> </tr> </table> </td></tr> </table> </div> ====Partial Volume #III-5==== This is the sub-volume that is painted black and identified as Partial Volume #5 (PV#5) in the right-hand panel of [[#ZoneIII|Figure 7]]. <div align="center"> <table border="1" cellpadding="5" align="center"> <tr><td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Lambda_5 </math> </td> <td align="center"> <math>~= </math> </td> <td align="left"> <math>~\infty</math> </td> </tr> <tr> <td align="right"> <math>~\lambda_5 </math> </td> <td align="center"> <math>~= </math> </td> <td align="left"> <math>~\xi_1|_\mathrm{max} = \biggl[1-\biggl( \frac{a}{\varpi_t-\beta_+} \biggr)^2 \biggr]^{-1/2} \approx 8.9243615</math> </td> </tr> </table> </td> <td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Gamma_{5} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ +1 </math> </td> </tr> <tr> <td align="right"> <math>~\gamma_{5} </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ -1 </math> </td> </tr> </table> </td></tr> </table> </div> <!-- HIDE SEGMENT Also, <div align="center" id="Parameters"> <table border="1" cellpadding="5" align="center"> <tr><td align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\Gamma_1 = \xi_2\biggr|_+</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \xi_1 - \frac{ (\xi_1^2-1)^{1/2}}{(\varpi_i/a)_+} </math> </td> </tr> <tr> <td align="right"> <math>~\gamma_1 = \xi_2\biggr|_-</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \xi_1 - \frac{ (\xi_1^2-1)^{1/2}}{(\varpi_i/a)_-} </math> </td> </tr> </table> </td></tr> </table> </div> where, in addition to the quantities defined above, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\biggl(\frac{\varpi_i}{a}\biggr)_\pm</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~\frac{\kappa}{2a^2}\cdot \frac{\mathrm{B}}{\mathrm{A}} \biggl[1 \pm \sqrt{1-\frac{AC}{B^2}} \biggr] </math> </td> </tr> <tr> <td align="right"> <math>~\mathrm{A}</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~\biggl(\frac{Z_0}{a}\biggr)^2 + \biggl[\frac{\varpi_t}{a} - \frac{\xi_1}{(\xi_1^2-1)^{1/2}}\biggr]^2 </math> </td> </tr> <tr> <td align="right"> <math>~\mathrm{B}</math> </td> <td align="center"> <math>~\equiv</math> </td> <td align="left"> <math>~\biggl(\frac{2\varpi_t Z_0^2}{a\kappa}\biggr) - \biggl[\frac{\varpi_t}{a} - \frac{\xi_1}{(\xi_1^2-1)^{1/2}}\biggr] </math> </td> </tr> </table> END HIDE SEGMENT --> '''Alternative determination of this partial volume:''' Notice that this "black" meridional-plane segment is a (semi)circle whose cross-sectional radius is (see [[Appendix/Ramblings/ToroidalCoordinates#Walk_Through_Step-By-Step|accompanying discussion]]), <div align="center"> <math>~r_\mathrm{black} = \frac{a}{\sqrt{\lambda_5^2 - 1}} \, ,</math> </div> and it is associated with a circular torus whose major radius is, <div align="center"> <math>~R_\mathrm{black} = \lambda_5 r_\mathrm{black} \, .</math> </div> Hence, using the [[2DStructure/ToroidalCoordinates#Total_Mass|familiar expression for the volume of a circular torus]], we know that the volume associated with this "black" partial volume is, <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~V_\mathrm{black}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{1}{2}\biggl[ 2\pi^2 R_\mathrm{black} r_\mathrm{black}^2 \biggr] = \pi^2 \lambda_5 r_\mathrm{black}^3 = \frac{\pi^2 a^3 \lambda_5}{(\lambda_3^2-1)^{3/2}} </math> </td> </tr> <tr> <td align="right"> <math>~\Rightarrow~~~~\frac{V_\mathrm{black}}{V_\mathrm{torus}}</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\biggl(\frac{a^3}{2\varpi_t r_t^2}\biggr) \frac{\lambda_5}{(\lambda_5^2-1)^{3/2}} \approx 0.06988365 \, .</math> </td> </tr> </table> </div> ====Summary (Zone III)==== <div align="center"> <table border="1" cellpadding="5" align="center"> <tr> <th align="center" colspan="7"><font size="+1">Table 2a:</font> Validate Pattern III-A</th> </tr> <tr><th align="center" colspan="6"> Integration Limits for model parameters <p></p><math>~(a, Z_0, \varpi_t, r_t) = (\tfrac{4}{5}, \tfrac{3}{20}, \tfrac{3}{4}, \tfrac{1}{4})</math> </th> <td align="center" rowspan="7">[[File:TorusTest000Sm.png|250px|Torus Test 000]]</td> </tr> <tr> <td align="center" colspan="1" rowspan="2"> </td> <td align="center"><math>~\xi_1|_\mathrm{max}</math></td> <td align="center"><math>~\xi_1|_\mathrm{min}</math></td> <td align="center"><math>~\xi_1|_+</math></td> <td align="center"><math>~\xi_1|_-</math></td> <td align="center" rowspan="2"><math>~\infty</math></td> </tr> <tr> <td align="center">8.92436</td> <td align="center">1.91162</td> <td align="center">5.87619</td> <td align="center">2.79259</td> </tr> <!-- TEMPORARY <tr> <td align="center">Temporary</td> <td align="center"><b><font color="pink">END</font></b><p></p><b><font color="red">END</font></b></td> <td align="center"><b><font color="blue">START</font></b></td> <td align="center"><b><font color="green">END</font></b><p></p><b><font color="red">START</font></b></td> <td align="center"> <b><font color="blue">END</font></b><p></p><b><font color="green">START</font></b><p></p><b><font color="pink">START</font></b> </td> <td align="center"> </td> </tr> END TEMPORARY--> <tr> <td align="center"><math>~\xi_2|_+</math></td> <td align="center"><b><font color="red">START</font></b></td> <td align="center"><b><font color="blue">END</font></b></td> <td align="center"> </td> <td align="center"><b><font color="green">END</font></b></td> <td align="center"> </td> </tr> <tr> <td align="center"><math>~\xi_2|_-</math></td> <td align="center"> </td> <td align="center"> </td> <td align="center"> </td> <td align="center"><b><font color="blue">START</font></b><p></p><b><font color="pink">END</font></b></td> <td align="center"> </td> </tr> <tr> <td align="center"><math>~+1</math></td> <td align="center"> </td> <td align="center"> </td> <td align="center"><b><font color="red">END</font></b></td> <td align="center"> </td> <td align="center"> </td> </tr> <tr> <td align="center"><math>~-1</math></td> <td align="center"><b><font color="pink">START</font></b></td> <td align="center"> </td> <td align="center"><b><font color="green">START</font></b></td> <td align="center"> </td> <td align="center"> </td> </tr> </table> </div> <!-- BEGIN OMIT <table border="1" cellpadding="8" align="center"> <tr><th align="center" colspan="13"> <font size="+1">Table 2:</font> Zone III Partial Volumes & Integration Limits on <math>~\xi_1</math><p></p> for model parameters <math>~(a, Z_0, \varpi_t, r_t) = (\tfrac{4}{5}, \tfrac{3}{20}, \tfrac{3}{4}, \tfrac{1}{4})</math> </th></tr> <tr> <td align="center" colspan="1"> </td> <td align="center" colspan="2">PV #1<p></p>(blue)</td> <td align="center" colspan="2">PV #2<p></p>(green)</td> <td align="center" colspan="2">PV #3<p></p>(orange)</td> <td align="center" colspan="4">PV #4<p></p>(pink) and (red)</td> <td align="center" colspan="2">PV #5<p></p>(black)</td> </tr> <tr> <td align="center" rowspan="2" colspan="1">Integration<p></p>Limits</td> <td align="center"><math>~\lambda_1</math></td> <td align="center"><math>~\Lambda_1</math></td> <td align="center"><math>~\lambda_2</math></td> <td align="center"><math>~\Lambda_2</math></td> <td align="center"><math>~\lambda_3</math></td> <td align="center"><math>~\Lambda_3</math></td> <td align="center"><math>~\lambda_{4a}</math></td> <td align="center"><math>~\Lambda_{4a}</math></td> <td align="center"><math>~\lambda_{4b}</math></td> <td align="center"><math>~\Lambda_{4b}</math></td> <td align="center"><math>~\lambda_5</math></td> <td align="center"><math>~\Lambda_5</math></td> </tr> <tr> <td align="center"><math>2.792593</math></td> <td align="center"><math>1.911617</math></td> <td align="center"><math>5.876190</math></td> <td align="center"><math>2.792593</math></td> <td align="center"><math>\infty</math></td> <td align="center"><math>5.876190</math></td> <td align="center"><math>8.924361</math></td> <td align="center"><math>2.792593</math></td> <td align="center"><math>8.924361</math></td> <td align="center"><math>5.876190</math></td> <td align="center"><math>\infty</math></td> <td align="center"><math>8.924361</math></td> </tr> <tr> <td align="center" rowspan="1" colspan="1">Volume<p></p>Fraction</td> <td align="center" colspan="2"><math>0.25309351</math></td> <td align="center" colspan="2"><math>0.43923615</math></td> <td align="center" colspan="2"><math>0.165291952</math></td> <td align="center" colspan="2"><math>0.059240412</math></td> <td align="center" colspan="2"><math>0.013254438</math></td> <td align="center" colspan="2"><math>0.0698836475</math></td> </tr> <tr> <td align="center" rowspan="1" colspan="1">Total Volume<p></p><font size="-1">(nzones = 5000)</font></td> <td align="center" colspan="12">1.000000110 <math>~\Rightarrow~</math> Error = -1.1E-7 </td> </tr> </table> END OMIT --> <div align="center"> <table border="1" cellpadding="5" align="center"> <tr> <th align="center" colspan="7"><font size="+1">Table 2b:</font> Validate Pattern III-A</th> </tr> <tr><th align="center" colspan="6"> Integration Limits for model parameters <p></p><math>~(a, Z_0, \varpi_t, r_t) = (\tfrac{4}{5}, \tfrac{1}{8}, \tfrac{3}{4}, \tfrac{1}{4})</math> </th> <td align="center" rowspan="7">[[File:TorusTest001Sm.png|250px|Torus Test 000]]</td> </tr> <tr> <td align="center" colspan="1" rowspan="2"> </td> <td align="center"><math>~\xi_1|_\mathrm{max}</math></td> <td align="center"><math>~\xi_1|_\mathrm{min}</math></td> <td align="center"><math>~\xi_1|_+</math></td> <td align="center"><math>~\xi_1|_-</math></td> <td align="center" rowspan="2"><math>~\infty</math></td> </tr> <tr> <td align="center">7.19569</td> <td align="center">1.98820</td> <td align="center">5.35175</td> <td align="center">2.60173</td> </tr> <!-- TEMPORARY <tr> <td align="center">Temporary</td> <td align="center"><b><font color="pink">END</font></b><p></p><b><font color="red">END</font></b></td> <td align="center"><b><font color="blue">START</font></b></td> <td align="center"><b><font color="green">END</font></b><p></p><b><font color="red">START</font></b></td> <td align="center"> <b><font color="blue">END</font></b><p></p><b><font color="green">START</font></b><p></p><b><font color="pink">START</font></b> </td> <td align="center"> </td> </tr> END TEMPORARY--> <tr> <td align="center"><math>~\xi_2|_+</math></td> <td align="center"><b><font color="red">START</font></b></td> <td align="center"><b><font color="blue">END</font></b></td> <td align="center"> </td> <td align="center"><b><font color="green">END</font></b></td> <td align="center"> </td> </tr> <tr> <td align="center"><math>~\xi_2|_-</math></td> <td align="center"> </td> <td align="center"> </td> <td align="center"> </td> <td align="center"><b><font color="blue">START</font></b><p></p><b><font color="pink">END</font></b></td> <td align="center"> </td> </tr> <tr> <td align="center"><math>~+1</math></td> <td align="center"> </td> <td align="center"> </td> <td align="center"><b><font color="red">END</font></b></td> <td align="center"> </td> <td align="center"> </td> </tr> <tr> <td align="center"><math>~-1</math></td> <td align="center"><b><font color="pink">START</font></b></td> <td align="center"> </td> <td align="center"><b><font color="green">START</font></b></td> <td align="center"> </td> <td align="center"> </td> </tr> </table> </div> <div align="center"> <table border="1" cellpadding="5" align="center"> <tr> <th align="center" colspan="7"><font size="+1">Table 2c:</font> Validate Pattern III-A</th> </tr> <tr><th align="center" colspan="6"> Integration Limits for model parameters <p></p><math>~(a, Z_0, \varpi_t, r_t) = (\tfrac{9}{10}, \tfrac{3}{40}, \tfrac{3}{4}, \tfrac{1}{4})</math> </th> <td align="center" rowspan="7">[[File:TorusTest003Sm.png|250px|Torus Test 000]]</td> </tr> <tr> <td align="center" colspan="1" rowspan="2"> </td> <td align="center"><math>~\xi_1|_\mathrm{max}</math></td> <td align="center"><math>~\xi_1|_\mathrm{min}</math></td> <td align="center"><math>~\xi_1|_+</math></td> <td align="center"><math>~\xi_1|_-</math></td> <td align="center" rowspan="2"><math>~\infty</math></td> </tr> <tr> <td align="center">11.4106</td> <td align="center">1.84912</td> <td align="center">10.6947</td> <td align="center">1.95431</td> </tr> <!-- TEMPORARY <tr> <td align="center">Temporary</td> <td align="center"><b><font color="pink">END</font></b><p></p><b><font color="red">END</font></b></td> <td align="center"><b><font color="blue">START</font></b></td> <td align="center"><b><font color="green">END</font></b><p></p><b><font color="red">START</font></b></td> <td align="center"> <b><font color="blue">END</font></b><p></p><b><font color="green">START</font></b><p></p><b><font color="pink">START</font></b> </td> <td align="center"> </td> </tr> END TEMPORARY--> <tr> <td align="center"><math>~\xi_2|_+</math></td> <td align="center"><b><font color="red">START</font></b></td> <td align="center"><b><font color="blue">END</font></b></td> <td align="center"> </td> <td align="center"><b><font color="green">END</font></b></td> <td align="center"> </td> </tr> <tr> <td align="center"><math>~\xi_2|_-</math></td> <td align="center"> </td> <td align="center"> </td> <td align="center"> </td> <td align="center"><b><font color="blue">START</font></b><p></p><b><font color="pink">END</font></b></td> <td align="center"> </td> </tr> <tr> <td align="center"><math>~+1</math></td> <td align="center"> </td> <td align="center"> </td> <td align="center"><b><font color="red">END</font></b></td> <td align="center"> </td> <td align="center"> </td> </tr> <tr> <td align="center"><math>~-1</math></td> <td align="center"><b><font color="pink">START</font></b></td> <td align="center"> </td> <td align="center"><b><font color="green">START</font></b></td> <td align="center"> </td> <td align="center"> </td> </tr> </table> </div> <div align="center"> <table border="1" cellpadding="5" align="center"> <tr> <th align="center" colspan="7"><font size="+1">Table 3a:</font> Validate Pattern III-B</th> </tr> <tr><th align="center" colspan="6"> Integration Limits for model parameters <p></p><math>~(a, Z_0, \varpi_t, r_t) = (\tfrac{13}{20}, \tfrac{1}{10}, \tfrac{3}{4}, \tfrac{1}{4})</math> </th> <td align="center" rowspan="7">[[File:TorusTest002BSm.png|250px|Torus Test 000]]</td> </tr> <tr> <td align="center" colspan="1" rowspan="2"> </td> <td align="center"><math>~\xi_1|_\mathrm{max}</math></td> <td align="center"><math>~\xi_1|_\mathrm{min}</math></td> <td align="center"><math>~\xi_1|_+</math></td> <td align="center"><math>~\xi_1|_-</math></td> <td align="center" rowspan="2"><math>~\infty</math></td> </tr> <tr> <td align="center">2.10621</td> <td align="center">5.71122</td> <td align="center">2.57592</td> <td align="center">4.58888</td> </tr> <tr> <td align="center">Temporary</td> <td align="center"><b><font color="BLUE">END</font></b></td> <td align="center"><b><font color="pink">START</font></b><p></p><b><font color="red">START</font></b></td> <td align="center"><b><font color="red">END</font></b><p></p><b><font color="GREEN">END</font></b><p></p><b><font color="BLUE">START</font></b></td> <td align="center"> <b><font color="pink">END</font></b><p></p><b><font color="green">START</font></b> </td> <td align="center"> </td> </tr> <tr> <td align="center"><math>~\xi_2|_+</math></td> <td align="center"><b><font color="BLUE">END</font></b></td> <td align="center"><b><font color="RED">START</font></b></td> <td align="center"> </td> <td align="center"> </td> <td align="center"> </td> </tr> <tr> <td align="center"><math>~\xi_2|_-</math></td> <td align="center"> </td> <td align="center"> </td> <td align="center"><b><font color="BLUE">START</font></b></td> <td align="center"> <b><font color="PINK">END</font></b><p></p><b><font color="green">START</font></b> </td> <td align="center"> </td> </tr> <tr> <td align="center"><math>~+1</math></td> <td align="center"> </td> <td align="center"> </td> <td align="center"><b><font color="GREEN">END</font></b><p></p><b><font color="RED">END</font></b></td> <td align="center"> </td> <td align="center"> </td> </tr> <tr> <td align="center"><math>~-1</math></td> <td align="center"> </td> <td align="center"> <b><font color="PINK">START</font></b> </td> <td align="center"> </td> <td align="center"> </td> <td align="center"> </td> </tr> </table> </div> <table border="1" cellpadding="8" align="center"> <tr><th align="center" colspan="11"> <font size="+1">Table 4:</font> Various Zone III Partial Volumes<p></p> for model parameters <math>~(\varpi_t, r_t) = (\tfrac{3}{4}, \tfrac{1}{4})</math> </th></tr> <tr> <td align="center"><math>~a</math></td> <td align="center"><math>~Z_0</math></td> <td align="center">nzones</td> <td align="center">PV#1<p></p>(blue)</td> <td align="center">PV#2<p></p>(green)</td> <td align="center">PV#3<p></p>(orange)</td> <td align="center">PV#4a<p></p>(pink)</td> <td align="center">PV#4b<p></p>(red)</td> <td align="center">PV#5<p></p>(black)</td> <td align="center">Error</td> <td align="center">Thumbnail</td> </tr> <tr> <td align="center">0.8</td> <td align="center">0.15</td> <td align="center">5000</td> <td align="center" colspan="6">(see Table 2 for details)</td> <td align="center">-1.1E-7</td> <td align="center">[[File:TorusTest000Sm.png|75px|Torus Test 000]] </tr> <tr> <td align="center"> </td> <td align="center">0.10</td> <td align="center">5000</td> <td align="center">0.075401233</td> <td align="center">0.44164551</td> <td align="center">0.230637507</td> <td align="center">0.088778341</td> <td align="center">0.010790998</td> <td align="center">0.152746462</td> <td align="center">-4.8E-8</td> <td align="center"> </td> </tr> <tr> <td align="center"> </td> <td align="center">0.125</td> <td align="center">5000</td> <td align="center">0.14811529</td> <td align="center">0.45526220</td> <td align="center">0.201121481</td> <td align="center">0.073678658</td> <td align="center">0.013215276</td> <td align="center">0.108607177</td> <td align="center">-8.7E-8</td> <td align="center">[[File:TorusTest001Sm.png|75px|Torus Test 001]]</td> </tr> <tr> <td align="center">0.65</td> <td align="center">0.1</td> <td align="center">5000</td> <td align="center">0.21505365</td> <td align="center">0.38298966</td> <td align="center">0.149641005</td> <td align="center">0.0028992093</td> <td align="center">0.15531546</td> <td align="center">0.0941011094</td> <td align="center">-9.6E-8</td> <td align="center">[[File:TorusTest002BSm.png|75px|Torus Test 002]]</td> </tr> <tr> <td align="center">0.9</td> <td align="center">0.075</td> <td align="center">5000</td> <td align="center">0.013247868</td> <td align="center">0.60594555</td> <td align="center">0.0688878257</td> <td align="center">0.25103223</td> <td align="center">0.00046944204</td> <td align="center">0.0604172907</td> <td align="center">-2.0E-7</td> <td align="center">[[File:TorusTest003Sm.png|75px|Torus Test 003]]</td> </tr> </table>
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