User:Tohline/Appendix/Ramblings/ConcentricEllipsodalT8Coordinates - Revision history

Tohline: /* Angle Between Unit Vectors */

2021-02-24T20:52:15Z

Angle Between Unit Vectors

← Older revision		Revision as of 20:52, 24 February 2021
Line 1,883:		Line 1,883:
	<math>~		<math>~
	\biggl[ \frac{p^2 z}{\mathcal{D} (x^2 + q^4y^2)^{1 / 2}} \biggr] \biggl[ x^2 - q^4y^2 \biggr] \, .		\biggl[ \frac{p^2 z}{\mathcal{D} (x^2 + q^4y^2)^{1 / 2}} \biggr] \biggl[ x^2 - q^4y^2 \biggr] \, .
			</math>
			</td>
			</tr>
			</table>


			----


			Let's again visit the unit vector that we know lies in the tangent-plane and is always orthogonal to <math>~\lambda_2</math>, namely,

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

			<tr>
			<td align="right">
			<math>~\hat{e}_{\lambda_3}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			-~\hat\imath \biggl[ x(2q^4 y^2 + p^4z^2 )\biggr]\frac{\ell_{3D}}{\mathcal{D}}
			+
			\hat\jmath \biggl[q^2y(p^4z^2 + 2x^2) \biggr]\frac{\ell_{3D}}{\mathcal{D}}
			+
			\hat{k} \biggl[p^2z (x^2 - q^4y^2) \biggr]\frac{\ell_{3D}}{\mathcal{D}}
			\, .
			</math>
			</td>
			</tr>
			</table>

			We acknowledge that,

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

			<tr>
			<td align="right">
			<math>~\hat{e}_{\lambda_2} \cdot \hat{e}_{\lambda_3}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl\{
			\hat\imath \biggl[ \frac{q^2y p^2z}{\mathcal{D}} \biggr]
			+
			\hat\jmath \biggl[ \frac{xp^2z}{\mathcal{D}} \biggr]
			-
			\hat{k} \biggl[ \frac{2x q^2y }{\mathcal{D}} \biggr]
			\biggr\}
			\biggl\{
			-~\hat\imath \biggl[ x(2q^4 y^2 + p^4z^2 )\biggr]\frac{\ell_{3D}}{\mathcal{D}}
			+
			\hat\jmath \biggl[q^2y(p^4z^2 + 2x^2) \biggr]\frac{\ell_{3D}}{\mathcal{D}}
			+
			\hat{k} \biggl[p^2z (x^2 - q^4y^2) \biggr]\frac{\ell_{3D}}{\mathcal{D}}
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

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

			<tr>
			<td align="right">

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

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			0 \, .
	</math>		</math>
	</td>		</td>

Tohline: /* Angle Between Unit Vectors */

2021-02-24T20:17:38Z

Angle Between Unit Vectors

← Older revision		Revision as of 20:17, 24 February 2021
Line 1,824:		Line 1,824:

	</td></tr></table>		</td></tr></table>
	From this pair of orthogonality checks, we appreciate that both unit vectors always lie in the plane that is tangent to the surface of our ellipsoid. Next, let's determine the angle, <math>~\alpha</math>, between these two unit vectors.

			From this pair of orthogonality checks, we appreciate that both unit vectors always lie in the plane that is tangent to the surface of our ellipsoid. Next, let's determine the angle, <math>~\alpha</math>, between these two unit vectors as measured in the relevant tangent-plane.

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

Tohline: /* Angle Between Unit Vectors */

2021-02-24T20:10:14Z

Angle Between Unit Vectors

← Older revision		Revision as of 20:10, 24 February 2021
Line 1,824:		Line 1,824:

	</td></tr></table>		</td></tr></table>
			From this pair of orthogonality checks, we appreciate that both unit vectors always lie in the plane that is tangent to the surface of our ellipsoid. Next, let's determine the angle, <math>~\alpha</math>, between these two unit vectors.

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

			<tr>
			<td align="right">
			<math>~\cos\alpha \equiv \hat{e}_{\lambda_2} \cdot \hat{e}_{\kappa_2}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl\{
			\hat\imath \biggl[ \frac{q^2y p^2z}{\mathcal{D}} \biggr]
			+
			\hat\jmath \biggl[ \frac{xp^2z}{\mathcal{D}} \biggr]
			-
			\hat{k} \biggl[ \frac{2x q^2y }{\mathcal{D}} \biggr]
			\biggr\}
			\biggl\{
			-\hat\imath \biggl[ \frac{q^2y}{(x^2 + q^4y^2)^{1 / 2}}\biggr]
			+
			\hat\jmath \biggl[ \frac{x}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
			+
			\hat{k} \biggl[ 0 \biggr]
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

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

			<tr>
			<td align="right">

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

	=Kappa (κ8) Coordinates=		=Kappa (κ8) Coordinates=

Tohline: /* Angle Between Unit Vectors */

2021-02-24T19:52:32Z

Angle Between Unit Vectors

← Older revision		Revision as of 19:52, 24 February 2021
Line 1,622:		Line 1,622:
	</tr>		</tr>
	</table>		</table>


			<table border="1" width="80%" align="center" cellpadding="8"><tr><td align="left">
			With regard to orthogonality, note that,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\hat{e}_1 \cdot \hat{e}_{\lambda_2}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl[
			\hat\imath (x\ell_{3D} )
			+
			\hat\jmath (q^2 y\ell_{3D} )
			+
			\hat{k} (p^2z\ell_{3D} )
			\biggr] \cdot
			\biggl\{
			\hat\imath \biggl[ \frac{q^2y p^2z}{\mathcal{D}} \biggr]
			+
			\hat\jmath \biggl[ \frac{xp^2z}{\mathcal{D}} \biggr]
			-
			\hat{k} \biggl[ \frac{2x q^2y }{\mathcal{D}} \biggr]
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

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

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			0 \, .</math>
			</td>
			</tr>
			</table>

			</td></tr></table>


	And,		And,
Line 1,695:		Line 1,762:
	</tr>		</tr>
	</table>		</table>


			<table border="1" width="80%" align="center" cellpadding="8"><tr><td align="left">
			Again, note that with regard to orthogonality,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\hat{e}_1 \cdot \hat{e}_{\kappa_2}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl[
			\hat\imath (x\ell_{3D} )
			+
			\hat\jmath (q^2 y\ell_{3D} )
			+
			\hat{k} (p^2z\ell_{3D} )
			\biggr] \cdot
			\biggl\{
			-\hat\imath \biggl[ \frac{q^2y}{(x^2 + q^4y^2)^{1 / 2}}\biggr]
			+
			\hat\jmath \biggl[ \frac{x}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			-~(x\ell_{3D} )\biggl[ \frac{q^2y}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
			+
			(q^2 y\ell_{3D} )\biggl[ \frac{x}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			0 \, .</math>
			</td>
			</tr>
			</table>

			</td></tr></table>

	=Kappa (κ8) Coordinates=		=Kappa (κ8) Coordinates=

Tohline: /* Angle Between Unit Vectors */

2021-02-24T18:48:30Z

Angle Between Unit Vectors

← Older revision		Revision as of 18:48, 24 February 2021
Line 1,673:		Line 1,673:
	<math>~		<math>~
	\frac{xq^2y}{\kappa_2 (x^2 + q^4y^2)^{1 / 2}}		\frac{xq^2y}{\kappa_2 (x^2 + q^4y^2)^{1 / 2}}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow~~~\hat{e}_{\kappa_2}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			-~\hat\imath \biggl[ \frac{q^2y}{(x^2 + q^4y^2)^{1 / 2}}\biggr]
			+
			\hat\jmath \biggl[ \frac{x}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
			+
			\hat{k} \biggl[ 0 \biggr] \, .
	</math>		</math>
	</td>		</td>

Tohline: /* Angle Between Unit Vectors */

2021-02-24T18:43:00Z

Angle Between Unit Vectors

Show changes

Tohline at 17:48, 24 February 2021

2021-02-24T17:48:35Z

← Older revision		Revision as of 17:48, 24 February 2021
Line 1,444:		Line 1,444:
	</tr>		</tr>
	</table>		</table>

			=Angle Between Unit Vectors=
			In the [[#TableKappa8\|Table below titled, "Direction Cosines Components for κ8 Coordinates"]], there are two fully-formed unit vectors that are each orthogonal to the (first) unit vector that is normal to the ellipsoid's surface. Here we will refer to the coordinates of these two "perpendicular" unit vectors as,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\lambda_2</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~\frac{xy^{1/q^2}}{z^{2/p^2}}</math>
			</td>
			<td align="center">      and,       </td>
			<td align="right">
			<math>~\kappa_2</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~\frac{(qy)^{1/q^2}}{x} \, .</math>
			</td>
			</tr>
			</table>



	=Kappa (κ8) Coordinates=		=Kappa (κ8) Coordinates=
Line 1,621:		Line 1,650:
	</table>		</table>


			<span id="TableKappa8"> </span>
	<table border="1" cellpadding="8" align="center">		<table border="1" cellpadding="8" align="center">
	<tr>		<tr>

Tohline: /* Blue-Framed Guess 2C */

2021-02-19T18:14:20Z

Blue-Framed Guess 2C

← Older revision		Revision as of 18:14, 19 February 2021
Line 2,777:		Line 2,777:
	</tr>		</tr>
	<tr>		<tr>
	<td align="center"><math>~n</math></td>		<td align="center"><math>~n</math><br />  <br />(1)</td>
	<td align="center"><math>~\lambda_n</math></td>		<td align="center"><math>~\lambda_n</math><br />  <br />(2)</td>
	<td align="center"><math>~h_n</math></td>		<td align="center"><math>~h_n</math><br />  <br />(3)</td>
	<td align="center"><math>~~~\frac{~~\partial \lambda_n}{\partial x}</math></td>		<td align="center"><math>~\partial \lambda_n/\partial x</math><br />  <br />(4)</td>
	<td align="center"><math>~~~\frac{~~\partial \lambda_n}{\partial y}</math></td>		<td align="center"><math>~\partial \lambda_n/\partial y</math><br />  <br />(5)</td>
	<td align="center"><math>~~~\frac{~~\partial \lambda_n}{\partial z}</math></td>		<td align="center"><math>~\partial \lambda_n/\partial z</math><br />  <br />(6)</td>
	<td align="center"><math>~\gamma_{n1}</math></td>		<td align="center"><math>~\gamma_{n1}</math><br />  <br />(7)</td>
	~~<td align="center"><math>~\gamma_{n2}~~</~~math~~></td>		<td align="center"><math>~\gamma_{n2}</math><br />  <br />(8)</td>
	<td align="center"><math>~\gamma_{n3}</math></td>		<td align="center"><math>~\gamma_{n3}</math><br />  <br />(9)</td>
	</~~tr>~~
	~~<tr>~~
	~~<td align="center"~~>(1)</td>
	<td align="center">~~(2)~~<~~/td~~>
	~~<td align="center">(3)~~</~~td>~~
	~~<td align="center"~~>~~(4)~~</td>
	~~<td align="center">(5)~~</~~td>~~
	~~<td align="center">(6)</td>~~
	~~<td align="center">(7)</td>~~
	~~<td align="center">(8)</td>~~
	~~<td align="center"~~>(9)</td>
	</tr>		</tr>

Tohline: /* Blue-Framed Guess 2C */

2021-02-19T18:07:56Z

Blue-Framed Guess 2C

← Older revision		Revision as of 18:07, 19 February 2021
Line 2,802:		Line 2,802:
	<td align="center"><math>~2A</math></td>		<td align="center"><math>~2A</math></td>
	<td align="center">---</td>		<td align="center">---</td>
	<td align="center" bgcolor="~~pink~~">~~<math>~\ell_{3D}(x^2 + q^4 y^2)^{1 / 2}</math>~~</td>		<td align="center" bgcolor="white">---</td>
	<td align="center" bgcolor="white">~~<math>~\frac{xp^2z}{(x^2 + q^4y^2)} </math>~~</td>		<td align="center" bgcolor="white">---</td>
	<td align="center" bgcolor="white">~~<math>~\frac{q^2 y p^2z}{(x^2 + q^4y^2)}</math>~~</td>		<td align="center" bgcolor="white">---</td>
	<td align="center" bgcolor="white">~~<math>~~~-~~1</math>~~</td>		<td align="center" bgcolor="white">---</td>
	<td align="center"><math>~~~\frac{~~x p^2 z\~~ell_{3D}}{(x^2 + q^4y^2)^~~{~~1 / 2}~~} </math></td>		<td align="center"><math>~x p^2 z ~\mathfrak{J}_0 </math></td>
	<td align="center"><math>~~~\frac{~~q^2 y p^2 z\~~ell_~~{~~3D}}{(x^2 + q^4y^2)^{1 / 2}~~} </math></td>		<td align="center"><math>~q^2 y p^2 z~\mathfrak{J}_0 </math></td>
	<td align="center"><math>~-~~\frac{~~(x^2 + q^4 y^2)\~~ell_{3D}}{(x^2 + q^4y^2)^~~{~~1 / 2}~~} </math></td>		<td align="center"><math>~-(x^2 + q^4 y^2)~\mathfrak{J}_0 </math></td>
	</tr>		</tr>

Tohline: /* Blue-Framed Guess 2C */

2021-02-18T21:06:08Z

Blue-Framed Guess 2C

← Older revision		Revision as of 21:06, 18 February 2021
Line 2,950:		Line 2,950:
	</tr>		</tr>
	</table>		</table>
	These cannot simultaneously be satisfied.		These cannot simultaneously be satisfied. So the blue-framed <font color="red">guess 2C does not provide a correct second-coordinate expression</font>. But we are very close! We need one additional scalar coefficient degree of freedom.

	===Next Thought===		===Next Thought===