User:Tohline/Appendix/Ramblings/ToroidalCoordinates - Revision history

Tohline: /* Examples */

2018-08-30T16:57:47Z

Examples

← Older revision		Revision as of 16:57, 30 August 2018
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	for a toroidal coordinate system whose origin (filled, red dot) remains fixed at the (cylindrical) coordinate location, <math>~(\varpi, z) = (a, Z_0) = (\tfrac{1}{3}, \tfrac{3}{4})</math>. For a toroidal coordinate system with this specified origin and an equatorial-plane torus having <math>~\varpi_t = \tfrac{3}{4}</math> and <math>~r_t = \tfrac{1}{4}</math> — as recorded in the top row of numbers in the Table, below — the <math>\xi_1</math>-circle makes ''first contact'' with the torus when <math>~\xi_1 = \xi_1\|_\mathrm{max} = 1.1927843</math> and it makes ''final contact'' when <math>~\xi_1 = \xi_1\|_\mathrm{min} = 1.0449467</math>. The animation sequence contains ten unique frames: The value of <math>~\xi_1</math> that is associated with the <math>\xi_1</math>-circle in each case appears near the bottom-right corner of the animation frame. These parameter values have also been recorded in the first column of ten separate rows in the following table, along with other relevant parameter values. For example, in each frame of the animation, the points of intersection between the surface of the torus and the <math>\xi_1</math>-circle are identified by filled, green diamonds; the (cylindrical) coordinates associated with these points of intersection, <math>~(\varpi_i, z_i)</math>, are listed in each table row, along with the corresponding value of the toroidal coordinate system's angular, <math>~\xi_2</math> coordinate.		for a toroidal coordinate system whose origin (filled, red dot) remains fixed at the (cylindrical) coordinate location, <math>~(\varpi, z) = (a, Z_0) = (\tfrac{1}{3}, \tfrac{3}{4})</math>. For a toroidal coordinate system with this specified origin and an equatorial-plane torus having <math>~\varpi_t = \tfrac{3}{4}</math> and <math>~r_t = \tfrac{1}{4}</math> — as recorded in the top row of numbers in the Table, below — the <math>\xi_1</math>-circle makes ''first contact'' with the torus when <math>~\xi_1 = \xi_1\|_\mathrm{max} = 1.1927843</math> and it makes ''final contact'' when <math>~\xi_1 = \xi_1\|_\mathrm{min} = 1.0449467</math>. The animation sequence contains ten unique frames: The value of <math>~\xi_1</math> that is associated with the <math>\xi_1</math>-circle in each case appears near the bottom-right corner of the animation frame. These parameter values have also been recorded in the first column of ten separate rows in the following table, along with other relevant parameter values. For example, in each frame of the animation, the points of intersection between the surface of the torus and the <math>\xi_1</math>-circle are identified by filled, green diamonds; the (cylindrical) coordinates associated with these points of intersection, <math>~(\varpi_i, z_i)</math>, are listed in each table row, along with the corresponding value of the toroidal coordinate system's angular, <math>~\xi_2</math> coordinate.

	<div id="Example2">		<div id="Example2" style="width: 85%; height: 15em; overflow: auto;">
	<table align="center" border="1" cellpadding="5">		<table align="center" border="1" cellpadding="5">
	<tr><th align="center" colspan="10">Example 2</th></tr>		<tr><th align="center" colspan="10">Example 2</th></tr>

Tohline: /* Advantageous Coordinate System */

2016-03-19T03:21:24Z

Advantageous Coordinate System

← Older revision		Revision as of 03:21, 19 March 2016
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	<td align="left">		<td align="left">
	<math>~(1-\xi_2^2)^{1/2} (\xi_1^2 - \xi_2^2)^{1/2} \biggl\{ (1-\xi_2^2) + (\xi_1^2 - \xi_2^2)		<math>~(1-\xi_2^2)^{1/2} (\xi_1^2 - \xi_2^2)^{1/2} \biggl\{ (1-\xi_2^2) + (\xi_1^2 - \xi_2^2)
	- ( \xi_1 - \xi_2 )(\xi_1^2 - \xi_2^2)^{1/2} \biggr\}^{-1} \, .		- ( \xi_1 - \xi_2 )(\xi_1^2 - \xi_2^2)^{1/2} \biggr\}^{-1}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl[ \frac{(1-\xi_2^2)}{ (\xi_1^2 - \xi_2^2) } \biggr]^{1/2} \biggl\{ \biggl[ \frac{(1-\xi_2^2)}{(\xi_1^2 - \xi_2^2)}\biggr] + 1
			- \biggl[ \frac{ ( \xi_1 - \xi_2 )}{(\xi_1 + \xi_2)} \biggr]^{1/2}\biggr\}^{-1} \, .
	</math>		</math>
	</td>		</td>

Tohline: /* Advantageous Coordinate System */ Insert links to Kojima and to PP84

2016-03-19T03:06:15Z

Advantageous Coordinate System: Insert links to Kojima and to PP84

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	===Advantageous Coordinate System===		===Advantageous Coordinate System===

	According to Kojima's (1986) review of the PP84 discussion — see his equation (14) — surfaces of constant density can be defined by the coordinate, <math>~\chi_{PP}</math>, where,		According to [http://adsabs.harvard.edu/abs/1986PThPh..75..251K Kojima's (1986)] review of the [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P PP84] discussion — see his equation (14) — surfaces of constant density can be defined by the coordinate, <math>~\chi_{PP}</math>, where,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</div>		</div>

	Indeed, equation (6.6) of PP84 defines the coordinate, <math>~\chi_{PP}</math>, via the expression,		Indeed, equation (6.6) of [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P PP84] defines the coordinate, <math>~\chi_{PP}</math>, via the expression,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Advantageous Coordinate System */

2016-03-19T03:03:29Z

Advantageous Coordinate System

← Older revision		Revision as of 03:03, 19 March 2016
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	<math>~(1-\xi_2^2)^{1/2} \biggl\{ \frac{(1-\xi_2^2)}{ (\xi_1 - \xi_2)} \biggl[ \frac{\xi_1 - \xi_2}{\xi_1 + \xi_2} \biggr]^{1/2}		<math>~(1-\xi_2^2)^{1/2} \biggl\{ \frac{(1-\xi_2^2)}{ (\xi_1 - \xi_2)} \biggl[ \frac{\xi_1 - \xi_2}{\xi_1 + \xi_2} \biggr]^{1/2}
	+ (\xi_1 + \xi_2)^{1/2} (\xi_1 - \xi_2)^{1/2} - ( \xi_1 - \xi_2 ) \biggr\}^{-1}		+ (\xi_1 + \xi_2)^{1/2} (\xi_1 - \xi_2)^{1/2} - ( \xi_1 - \xi_2 ) \biggr\}^{-1}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~(1-\xi_2^2)^{1/2} \biggl\{ \frac{(1-\xi_2^2)}{ (\xi_1 - \xi_2)^{1/2} (\xi_1 + \xi_2)^{1/2}}
			+ (\xi_1 + \xi_2)^{1/2} (\xi_1 - \xi_2)^{1/2} - ( \xi_1 - \xi_2 ) \biggr\}^{-1}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~(1-\xi_2^2)^{1/2} (\xi_1 - \xi_2)^{1/2} (\xi_1 + \xi_2)^{1/2} \biggl\{ (1-\xi_2^2) + (\xi_1 - \xi_2)(\xi_1 + \xi_2)
			- ( \xi_1 - \xi_2 )(\xi_1 + \xi_2)^{1/2} (\xi_1 - \xi_2)^{1/2} \biggr\}^{-1}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~(1-\xi_2^2)^{1/2} (\xi_1^2 - \xi_2^2)^{1/2} \biggl\{ (1-\xi_2^2) + (\xi_1^2 - \xi_2^2)
			- ( \xi_1 - \xi_2 )(\xi_1^2 - \xi_2^2)^{1/2} \biggr\}^{-1} \, .
	</math>		</math>
	</td>		</td>
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	</table>		</table>
	</div>		</div>
			This does not appear to be very useful or productive!

	==CCGF Expansion==		==CCGF Expansion==

Tohline: /* Advantageous Coordinate System */

2016-03-19T02:36:02Z

Advantageous Coordinate System

← Older revision		Revision as of 02:36, 19 March 2016
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	</table>		</table>
	</div>		</div>
	Hence, they are the same, as long as we appreciate that Kojima assumes all length scales are normalized to <math>~\varpi_0</math>.		Hence, they are the same, as long as we appreciate that Kojima assumes all length scales are normalized to <math>~\varpi_0</math>. Let's express this coordinate in terms of the <math>~(\xi_1, \xi_2)</math> toroidal coordinates as defined by MF53, namely,

			<table align="center" border="0" cellpadding="5">

			<tr>
			<td align="right">
			<math>
			~\frac{z}{a}
			</math>
			</td>
			<td align="center">
			<math>
			~=
			</math>
			</td>
			<td align="left">
			<math>
			~\frac{(1-\xi_2^2)^{1/2}}{\xi_1 - \xi_2} \, ,
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>
			~\frac{\varpi}{a} \equiv \biggl[ \biggl(\frac{x}{a}\biggr)^2 + \biggl(\frac{y}{a}\biggr)^2 \biggr]^{1/2}
			</math>
			</td>
			<td align="center">
			<math>
			~=
			</math>
			</td>
			<td align="left">
			<math>
			~\frac{(\xi_1^2 - 1)^{1/2}}{\xi_1 - \xi_2} \, ,
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>
			~\frac{r}{a} \equiv \biggl[ \biggl(\frac{x}{a}\biggr)^2 + \biggl(\frac{y}{a}\biggr)^2 + \biggl(\frac{z}{a}\biggr)^2\biggr]^{1/2}
			</math>
			</td>
			<td align="center">
			<math>
			~=
			</math>
			</td>
			<td align="left">
			<math>
			~\biggl[ \frac{\xi_1 + \xi_2}{\xi_1 - \xi_2} \biggr]^{1/2} \, .
			</math>
			</td>
			</tr>
			</table>

			Kojima's expression becomes:
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\tan\chi_{PP}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~z \biggl[ \frac{z^2}{\sqrt{\varpi^2 + z^2}} + \sqrt{\varpi^2 + z^2} - 1 \biggr]^{-1}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{(1-\xi_2^2)^{1/2}}{\xi_1 - \xi_2} \biggl\{ \biggl[\frac{(1-\xi_2^2)}{ (\xi_1 - \xi_2)^2} \biggr] \biggl[ \frac{\xi_1 - \xi_2}{\xi_1 + \xi_2} \biggr]^{1/2}
			+ \biggl[ \frac{\xi_1 + \xi_2}{\xi_1 - \xi_2} \biggr]^{1/2} - 1 \biggr\}^{-1}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~(1-\xi_2^2)^{1/2} \biggl\{ \frac{(1-\xi_2^2)}{ (\xi_1 - \xi_2)} \biggl[ \frac{\xi_1 - \xi_2}{\xi_1 + \xi_2} \biggr]^{1/2}
			+ (\xi_1 + \xi_2)^{1/2} (\xi_1 - \xi_2)^{1/2} - ( \xi_1 - \xi_2 ) \biggr\}^{-1}
			</math>
			</td>
			</tr>
			</table>
			</div>

	==CCGF Expansion==		==CCGF Expansion==

Tohline: /* Papaloizou-Pringle Tori */ Begin to examine PP84's curvilinear coordinate system.

2016-03-19T02:04:25Z

Papaloizou-Pringle Tori: Begin to examine PP84's curvilinear coordinate system.

← Older revision		Revision as of 02:04, 19 March 2016
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	</table>		</table>

			===Advantageous Coordinate System===

			According to Kojima's (1986) review of the PP84 discussion — see his equation (14) — surfaces of constant density can be defined by the coordinate, <math>~\chi_{PP}</math>, where,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\tan\chi_{PP}</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~
			\frac{z}{ \frac{z^2}{\sqrt{\varpi^2 + z^2}} + \sqrt{\varpi^2 + z^2} - 1}
			</math>
			</td>
			</tr>
			</table>
			</div>

			Indeed, equation (6.6) of PP84 defines the coordinate, <math>~\chi_{PP}</math>, via the expression,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\tan\chi_{PP}</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~
			\frac{\cos\theta}{ \cos^2\theta + 1 - \varpi_0/r} \, ,
			</math>
			</td>
			</tr>
			</table>
			</div>
			where,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~z = r\cos\theta</math>
			</td>
			<td align="center">
			and
			</td>
			<td align="left">
			<math>~\varpi = r\sin\theta \, .</math>
			</td>
			</tr>
			</table>
			</div>

			Let's see if these match. Starting from the Kojima expression, we have,

			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\tan\chi_{PP}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{r\cos\theta}{ \frac{r^2\cos^2\theta}{r} + r - 1}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{\cos\theta}{ \cos^2\theta + 1 - 1/r} \, .
			</math>
			</td>
			</tr>
			</table>
			</div>
			Hence, they are the same, as long as we appreciate that Kojima assumes all length scales are normalized to <math>~\varpi_0</math>.

	==CCGF Expansion==		==CCGF Expansion==

Tohline: /* Toroidal Coordinates */ Add "span ID" tag to section that details toroidal-coordinate scale factors

2015-10-06T19:51:02Z

Toroidal Coordinates: Add "span ID" tag to section that details toroidal-coordinate scale factors

← Older revision		Revision as of 19:51, 6 October 2015
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	</table>		</table>

	According to MF53, the associated scale factors of this orthogonal coordinate system are:		<span id="ToroidalScaleFactors">According to MF53, the associated scale factors of this orthogonal coordinate system are:</span>
	<table align="center" border="0" cellpadding="5">		<table align="center" border="0" cellpadding="5">
	<tr>		<tr>

Tohline: /* Toroidal Configurations and Related Coordinate Systems */ Provide link to summary chapter

2015-10-05T19:41:57Z

Toroidal Configurations and Related Coordinate Systems: Provide link to summary chapter

← Older revision		Revision as of 19:41, 5 October 2015
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	=Toroidal Configurations and Related Coordinate Systems=		=Toroidal Configurations and Related Coordinate Systems=

			This rather long and rambling chapter reveals how my train of thought progressed as I effectively taught myself how to use a toroidal coordinate system to determine the gravitational potential of an axisymmetric configuration. A summary of the key results from this chapter can be found in a primary chapter of this H_Book titled, [[User:Tohline/2DStructure/ToroidalCoordinates\|Using Toroidal Coordinates to Determine the Gravitational Potential]].


	{{LSU_HBook_header}}		{{LSU_HBook_header}}

Tohline: /* Examples */ A little more explanation of the animation figure

2015-10-05T03:07:39Z

Examples: A little more explanation of the animation figure

← Older revision		Revision as of 03:07, 5 October 2015
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	</div>		</div>

			Notice in the animation that, while the origin of the selected toroidal coordinate system (the filled red dot) remains fixed, the ''center'' of the <math>\xi_1</math>-circle does not remain fixed. In order to highlight this behavior, the location of the center of the <math>\xi_1</math>-circle has been marked by a filled, light-blue square and, in keeping with the earlier Figure 2 sketch, a vertical, light-blue line connects this center to the equatorial plane.

Tohline: /* Examples */ Write paragraph describing information contained in the animation sequence and Table

2015-10-05T02:56:27Z

Examples: Write paragraph describing information contained in the animation sequence and Table

← Older revision		Revision as of 02:56, 5 October 2015
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	===Examples===		===Examples===


			<div align="center" id="Figure2">
			<table border="1" cellpadding="8">

			<tr>
			<td align="center">
			[[File:DiagramToroidalCoordinates.png\|350px\|Diagram of Torus and Toroidal Coordinates]]
			</td>
			<td align="center">
			[[File:TCoordsE.gif\|Diagram of Torus and Toroidal Coordinates]]
			</td>
			</tr>
			</table>
			</div>

			For reference purposes, Figure 2 has been displayed here, again, in the lefthand panel of Figure 4; the animation sequence presented in the righthand panel illustrates how the <math>\xi_1</math>-circle (depicted by the locus of small black dots) intersects the surface of the (pink) equatorial-plane torus as the value of <math>~\xi_1</math> is varied over the parameter range,
			<div align="center">
			<math>~\xi_1\|_\mathrm{max} \geq \xi_1 \geq ~\xi_1\|_\mathrm{min} \, ,</math>
			</div>
			for a toroidal coordinate system whose origin (filled, red dot) remains fixed at the (cylindrical) coordinate location, <math>~(\varpi, z) = (a, Z_0) = (\tfrac{1}{3}, \tfrac{3}{4})</math>. For a toroidal coordinate system with this specified origin and an equatorial-plane torus having <math>~\varpi_t = \tfrac{3}{4}</math> and <math>~r_t = \tfrac{1}{4}</math> — as recorded in the top row of numbers in the Table, below — the <math>\xi_1</math>-circle makes ''first contact'' with the torus when <math>~\xi_1 = \xi_1\|_\mathrm{max} = 1.1927843</math> and it makes ''final contact'' when <math>~\xi_1 = \xi_1\|_\mathrm{min} = 1.0449467</math>. The animation sequence contains ten unique frames: The value of <math>~\xi_1</math> that is associated with the <math>\xi_1</math>-circle in each case appears near the bottom-right corner of the animation frame. These parameter values have also been recorded in the first column of ten separate rows in the following table, along with other relevant parameter values. For example, in each frame of the animation, the points of intersection between the surface of the torus and the <math>\xi_1</math>-circle are identified by filled, green diamonds; the (cylindrical) coordinates associated with these points of intersection, <math>~(\varpi_i, z_i)</math>, are listed in each table row, along with the corresponding value of the toroidal coordinate system's angular, <math>~\xi_2</math> coordinate.

	<div id="Example2">		<div id="Example2">
Line 2,776:		Line 2,797:
	<td align="center" colspan="1"><math>~-0.220722</math></td>		<td align="center" colspan="1"><math>~-0.220722</math></td>
	<td align="center" colspan="3">Degenerate Coordinate Values</td>		<td align="center" colspan="3">Degenerate Coordinate Values</td>
	~~</tr>~~
	~~</table>~~
	~~</div>~~


	~~<div align="center" id="Figure2">~~
	~~<table border="1" cellpadding="8">~~

	~~<tr>~~
	~~<td align="center">~~
	~~[[File:TCoordsE.gif\|Diagram of Torus and Toroidal Coordinates]]~~
	~~</td>~~
	</tr>		</tr>
	</table>		</table>