Appendix/Ramblings/T6CoordinatesPt2 - Revision history

Jet53man: /* Speculation7 */

2021-07-23T22:13:42Z

Speculation7

← Older revision		Revision as of 18:13, 23 July 2021
Line 3,698:		Line 3,698:

	Let's check this out.		Let's check this out.
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\mathrm{RHS}~\equiv \biggl(\frac{\mathcal{D}}{\ell_{3D}} \biggr)^2 \frac{1}{\ell_{3D}^4}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~(q^4 y^2 p^4 z^2 + x^2 p^4 z^2 + 4x^2q^4y^2)(x^2 + q^4y^2 + p^4 z^2 )^3</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~(x^2 + q^4y^2 + p^4 z^2 )(q^4 y^2 p^4 z^2 + x^2 p^4 z^2 + 4x^2q^4y^2)[x^4 + 2x^2q^4y^2 + 2x^2p^4z^2 +q^8y^4 + 2q^4y^2p^4z^2 + p^8z^4]</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			[(6x^2q^4 y^2 p^4 z^2 + x^4 p^4 z^2 + 4x^4q^4y^2)
			+
			(q^8 y^4 p^4 z^2 + 4x^2q^8 y^4)
			+
			(q^4 y^2 p^8 z^4 + x^2 p^8 z^4 )]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~\times [x^4 + 2x^2q^4y^2 + 2x^2p^4z^2 +q^8y^4 + 2q^4y^2p^4z^2 + p^8z^4]</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			[6x^2q^4 y^2 p^4 z^2 + x^4(p^4 z^2 + 4q^4y^2)
			+
			q^8 y^4(p^4 z^2 + 4x^2)
			+
			p^8 z^4 (q^4 y^2 + x^2 )]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~\times [x^4 + 2x^2q^4y^2 + 2x^2p^4z^2 +q^8y^4 + 2q^4y^2p^4z^2 + p^8z^4]</math>
			</td>
			</tr>
			</table>

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Jet53man: /* Speculation7 */

2021-07-23T22:12:42Z

Speculation7

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Jet53man: /* Speculation7 */

2021-07-23T22:12:05Z

Speculation7

@@ Line 3,506: / Line 3,506: @@
    <td align="center"><math>~\frac{\ell_{3D}}{\mathcal{D}}\biggl[ q^2 y(p^4z^2 + 2x^2 )  \biggl]</math></td>
    <td align="center"><math>~\frac{\ell_{3D}}{\mathcal{D}}\biggl[ p^2z( x^2 - q^4y^2 ) \biggl]</math></td>
 </tr>
 </table>
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Jet53man: /* Speculation7 */

2021-07-23T22:09:53Z

Speculation7

← Older revision		Revision as of 18:09, 23 July 2021
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	===Speculation7===		===Speculation7===

			<table border="1" cellpadding="8" align="center">
			<tr>
			<td align="center" colspan="9">'''Direction Cosine Components for T6 Coordinates'''</td>
			</tr>
			<tr>
			<td align="center"><math>~n</math></td>
			<td align="center"><math>~\lambda_n</math></td>
			<td align="center"><math>~h_n</math></td>
			<td align="center"><math>~\frac{\partial \lambda_n}{\partial x}</math></td>
			<td align="center"><math>~\frac{\partial \lambda_n}{\partial y}</math></td>
			<td align="center"><math>~\frac{\partial \lambda_n}{\partial z}</math></td>
			<td align="center"><math>~\gamma_{n1}</math></td>
			<td align="center"><math>~\gamma_{n2}</math></td>
			<td align="center"><math>~\gamma_{n3}</math></td>
			</tr>

			<tr>
			<td align="center"><math>~1</math></td>
			<td align="center"><math>~(x^2 + q^2 y^2 + p^2 z^2)^{1 / 2} </math></td>
			<td align="center"><math>~\lambda_1 \ell_{3D}</math></td>
			<td align="center"><math>~\frac{x}{\lambda_1}</math></td>
			<td align="center"><math>~\frac{q^2 y}{\lambda_1}</math></td>
			<td align="center"><math>~\frac{p^2 z}{\lambda_1}</math></td>
			<td align="center"><math>~(x) \ell_{3D}</math></td>
			<td align="center"><math>~(q^2 y)\ell_{3D}</math></td>
			<td align="center"><math>~(p^2z) \ell_{3D}</math></td>
			</tr>

			<tr>
			<td align="center"><math>~2</math></td>
			<td align="center"><math>~\frac{x y^{1/q^2}}{ z^{2/p^2}}</math></td>
			<td align="center"><math>~\frac{1}{\lambda_2}\biggl[\frac{x q^2 y p^2 z}{ \mathcal{D}}\biggr] </math></td>
			<td align="center"><math>~\frac{\lambda_2}{x}</math></td>
			<td align="center"><math>~\frac{\lambda_2}{q^2 y}</math></td>
			<td align="center"><math>~-\frac{2\lambda_2}{p^2 z}</math></td>
			<td align="center"><math>~\frac{x q^2 y p^2 z}{\mathcal{D}} \biggl(\frac{1}{x}\biggr)</math></td>
			<td align="center"><math>~ \frac{x q^2 y p^2 z}{\mathcal{D}} \biggl(\frac{1}{q^2y}\biggr)</math></td>
			<td align="center"><math>~\frac{x q^2 y p^2 z}{\mathcal{D}} \biggl(-\frac{2}{p^2z}\biggr)</math></td>
			</tr>

			<tr>
			<td align="center"><math>~3</math></td>
			<td align="center">---</td>
			<td align="center">---</td>
			<td align="center">---</td>
			<td align="center">---</td>
			<td align="center">---</td>
			<td align="center"><math>~-\frac{\ell_{3D}}{\mathcal{D}}\biggl[ x(2 q^4y^2 + p^4z^2 ) \biggl]</math></td>
			<td align="center"><math>~\frac{\ell_{3D}}{\mathcal{D}}\biggl[ q^2 y(p^4z^2 + 2x^2 ) \biggl]</math></td>
			<td align="center"><math>~\frac{\ell_{3D}}{\mathcal{D}}\biggl[ p^2z( x^2 - q^4y^2 ) \biggl]</math></td>
			</tr>
			</table>

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Jet53man: /* Orthogonal Coordinates */

2021-07-23T22:08:16Z

Orthogonal Coordinates

← Older revision		Revision as of 18:08, 23 July 2021
Line 3,453:		Line 3,453:

	No! Once again this does not work. The direction cosines — and, hence, the components of the <math>~\hat{e}_3</math> unit vector — are not correct!		No! Once again this does not work. The direction cosines — and, hence, the components of the <math>~\hat{e}_3</math> unit vector — are not correct!

			===Speculation7===

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Jet53man: /* Speculation6 */

2021-07-23T22:01:43Z

Speculation6

← Older revision		Revision as of 18:01, 23 July 2021
Line 3,367:		Line 3,367:
	And this is ''precisely'' what, just a few lines above, we hypothesized the functional expression for <math>~h_3</math> ought to be. <font color="red">'''EUREKA!'''</font>		And this is ''precisely'' what, just a few lines above, we hypothesized the functional expression for <math>~h_3</math> ought to be. <font color="red">'''EUREKA!'''</font>

			====Summary====

			In summary, then …
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			<tr>
			<td align="right">
			<math>~\lambda_3</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~
			-x^2 \biggl[ \mathfrak{F}_x \biggl]^{1 / 2}
			+ y^2 \biggl[ \mathfrak{F}_y \biggl]^{1 / 2}
			+ z^2 \biggl[ \mathfrak{F}_z \biggl]^{1 / 2}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			-x^2 \biggl[\biggl(q^4y^2 + \frac{p^4z^2}{2} \biggr)^2 \biggl]^{1 / 2}
			+ y^2 \biggl[ q^4\biggl( x^2 + \frac{p^4z^2}{2} \biggr)^2\biggl]^{1 / 2}
			+ z^2 \biggl[ \frac{p^4}{4} \biggl(x^2 - q^4y^2 \biggr)^2 \biggl]^{1 / 2}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			-x^2 \biggl(q^4y^2 + \frac{p^4z^2}{2} \biggr)
			+ q^2y^2 \biggl( x^2 + \frac{p^4z^2}{2} \biggr)
			+ \frac{p^2 z^2}{2} \biggl(x^2 - q^4y^2 \biggr) \, ,
			</math>
			</td>
			</tr>
			</table>
			and,


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

			<tr>
			<td align="right">
			<math>~h_3</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~(AB)^{-1 / 2} </math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl[
			(x^2 + q^4y^2 + p^4 z^2 )(q^4 y^2 p^4 z^2 + x^2 p^4 z^2 + 4x^2q^4y^2)
			\biggr]^{-1 / 2} \, .</math>
			</td>
			</tr>
			</table>

			No! Once again this does not work. The direction cosines — and, hence, the components of the <math>~\hat{e}_3</math> unit vector — are not correct!

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Jet53man at 21:53, 23 July 2021

2021-07-23T21:53:54Z

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Jet53man: /* Search for Third Coordinate Expression */

2021-07-23T21:51:50Z

Search for Third Coordinate Expression

@@ Line 2,883: / Line 2,883: @@
 <font color="red">'''Wow! &nbsp; Really close!''' (13 November 2020)</font>
-Just for fun, let's see what we get for <math>~h_3</math>.  It is given by the expression,
+<tr>
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Jet53man: /* Search for Third Coordinate Expression */

2021-07-23T21:50:34Z

Search for Third Coordinate Expression

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Jet53man: /* Search for Third Coordinate Expression */

2021-07-23T21:46:15Z

Search for Third Coordinate Expression

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