Apps/MaclaurinSpheroids/GoogleBooks - Revision history

Joel2: /* Prolate Spheroid */

2024-06-22T21:38:33Z

Prolate Spheroid

← Older revision		Revision as of 17:38, 22 June 2024
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	While trying to decipher Maclaurin's geometrically based derivation, we have found it useful to re-draw key elements of this figure, deriving relevant algebraic or trigonometric relations along the way. Because the "oblong" ellipse in Maclaurin's figure is oriented such that the axis of revolution — the symmetry axis — is horizontal, and because we prefer to associate the <math>~z</math> axis with the symmetry axis of the configuration, we will rather unconventionally refer to the horizontal axis as the <math>~z</math> (symmetry) axis. Adopting a cylindrical coordinate system furthermore means that, in our notation, the vertical axis will be the <math>~\varpi</math> axis.		While trying to decipher Maclaurin's geometrically based derivation, we have found it useful to re-draw key elements of this figure, deriving relevant algebraic or trigonometric relations along the way. Because the "oblong" ellipse in Maclaurin's figure is oriented such that the axis of revolution — the symmetry axis — is horizontal, and because we prefer to associate the <math>z</math> axis with the symmetry axis of the configuration, we will rather unconventionally refer to the horizontal axis as the <math>z</math> (symmetry) axis. Adopting a cylindrical coordinate system furthermore means that, in our notation, the vertical axis will be the <math>\varpi</math> axis.

	The blue ellipse centered on the origin of our plot represents Maclaurin's "ADBE" ellipse; in the expressions that follow, we will use <math>a_1</math> and <math>a_3</math> to refer, respectively, to the length of the semi-major axis (Maclaurin's line-segment "CA") and semi-minor axis (Maclaurin's line-segment "CD") of the ellipse; and, following Maclaurin, we will refer to the leftmost point on the ellipse (Maclaurin's point "A") as the "pole" which is positioned at coordinate <math>(z, \varpi) = (-a_1, 0)</math>. (Note that in our plot, we have set <math>a_1 = 1</math>, but otherwise we will leave its specification arbitrary.) In our plot, Maclaurin's "CNH" circle is represented by the green quarter circle centered on the pole. The blue ellipse is defined by the relation,		The blue ellipse centered on the origin of our plot represents Maclaurin's "ADBE" ellipse; in the expressions that follow, we will use <math>a_1</math> and <math>a_3</math> to refer, respectively, to the length of the semi-major axis (Maclaurin's line-segment "CA") and semi-minor axis (Maclaurin's line-segment "CD") of the ellipse; and, following Maclaurin, we will refer to the leftmost point on the ellipse (Maclaurin's point "A") as the "pole" which is positioned at coordinate <math>(z, \varpi) = (-a_1, 0)</math>. (Note that in our plot, we have set <math>a_1 = 1</math>, but otherwise we will leave its specification arbitrary.) In our plot, Maclaurin's "CNH" circle is represented by the green quarter circle centered on the pole. The blue ellipse is defined by the relation,
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\varpi</math>		<math>\varpi</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~(a_1+z)\tan\alpha</math>		<math>(a_1+z)\tan\alpha</math>
	</td>		</td>
	</tr>		</tr>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\Rightarrow ~~~~ \frac{\varpi}{a_1}</math>		<math>\Rightarrow ~~~~ \frac{\varpi}{a_1}</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\biggl(1+\frac{z}{a_1}\biggr)\tan\alpha \, .</math>		<math>\biggl(1+\frac{z}{a_1}\biggr)\tan\alpha \, .</math>
	</td>		</td>
	</tr>		</tr>

Joel2: /* Interpreting Maclaurin's Key Concluding Theorem */

2024-06-22T21:26:55Z

Interpreting Maclaurin's Key Concluding Theorem

← Older revision		Revision as of 17:26, 22 June 2024
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	Here we assess this ''geometrically'' formulated statement using today's more common differential operators and algebraic expressions. In particular, we draw from the [[Apps/MaclaurinSpheroids#Maclaurin_Spheroids_.28axisymmetric_structure.29\|discussion of Maclaurin spheroids that has been presented in an accompanying chapter]] of this H_Book.		Here we assess this ''geometrically'' formulated statement using today's more common differential operators and algebraic expressions. In particular, we draw from the [[Apps/MaclaurinSpheroids#Maclaurin_Spheroids_.28axisymmetric_structure.29\|discussion of Maclaurin spheroids that has been presented in an accompanying chapter]] of this H_Book.

	<font color="maroon">'''CA'''</font>: This is the semi-minor (polar) axis of the spheroid, which we have called, <math>~a_3</math>.		<font color="maroon">'''CA'''</font>: This is the semi-minor (polar) axis of the spheroid, which we have called, <math>a_3</math>.

	<font color="maroon">'''CD'''</font>: This is the semi-major (equatorial) radius of the spheroid, which we have called, <math>~a_1</math>.		<font color="maroon">'''CD'''</font>: This is the semi-major (equatorial) radius of the spheroid, which we have called, <math>a_1</math>.

	With both <math>a_1</math> and <math>a_3</math> specified, the eccentricity of the oblate spheroid is also known via the expression,		With both <math>a_1</math> and <math>a_3</math> specified, the eccentricity of the oblate spheroid is also known via the expression,
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	<td align="right">		<td align="right">
	</td>
			</td>
	<td align="center">		<td align="center">
	<math>=</math>		<math>=</math>
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	</tr>		</tr>
	</table>		</table>


	<font color="maroon">'''Centrifugal force in the equatorial plane arising from rotation'''</font>: We interpret this phrase to mean the centrifugal acceleration (force per unit mass) given by the expression,		<font color="maroon">'''Centrifugal force in the equatorial plane arising from rotation'''</font>: We interpret this phrase to mean the centrifugal acceleration (force per unit mass) given by the expression,
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	</tr>		</tr>
	</table>		</table>


	Maclaurin's theorem states that the rotating spheroidal configuration will be in equilibrium if,		Maclaurin's theorem states that the rotating spheroidal configuration will be in equilibrium if,
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	</tr>		</tr>
	</table>		</table>


	We choose to rewrite this expression and label it as,		We choose to rewrite this expression and label it as,

	<div align="center">		<div align="center" id="MaclaurinTheorem">
	~~<span~~ id="MaclaurinTheorem"><font color="#770000">'''Maclaurin's Theorem'''</font></span><br />		<font color="#770000">'''Maclaurin's Theorem'''</font></span><br />

	<math>\frac{\mathcal{V}}{a_1} = \frac{1}{a_1}\biggl[\mathcal{D} - \mathcal{A}\biggl( \frac{a_3}{a_1} \biggr)\biggr] </math>		<math>\frac{\mathcal{V}}{a_1} = \frac{1}{a_1}\biggl[\mathcal{D} - \mathcal{A}\biggl( \frac{a_3}{a_1} \biggr)\biggr] </math>

Joel2: /* Prolate Spheroid */

2024-06-22T21:20:59Z

Prolate Spheroid

← Older revision		Revision as of 17:20, 22 June 2024
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	<td align="left">		<td align="left">
	In this animation sequence, the blue ellipse has an eccentricity, <math>~e = 0.55</math>, and the red line-segment is drawn at an angle to the horizontal axis that varies over the range, <math>~\tfrac{3\pi}{8} \ge \alpha \ge \tfrac{\pi}{8}</math>. As is described in the accompanying text, the orange curve — Maclaurin's "AKG" curve — is mapped out as the angle, <math>~\alpha</math>, is varied.		In this animation sequence, the blue ellipse has an eccentricity, <math>e = 0.55</math>, and the red line-segment is drawn at an angle to the horizontal axis that varies over the range, <math>\tfrac{3\pi}{8} \ge \alpha \ge \tfrac{\pi}{8}</math>. As is described in the accompanying text, the orange curve — Maclaurin's "AKG" curve — is mapped out as the angle, <math>\alpha</math>, is varied.
	</td>		</td>
	</tr>		</tr>
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	While trying to decipher Maclaurin's geometrically based derivation, we have found it useful to re-draw key elements of this figure, deriving relevant algebraic or trigonometric relations along the way. Because the "oblong" ellipse in Maclaurin's figure is oriented such that the axis of revolution — the symmetry axis — is horizontal, and because we prefer to associate the <math>~z</math> axis with the symmetry axis of the configuration, we will rather unconventionally refer to the horizontal axis as the <math>~z</math> (symmetry) axis. Adopting a cylindrical coordinate system furthermore means that, in our notation, the vertical axis will be the <math>~\varpi</math> axis.		While trying to decipher Maclaurin's geometrically based derivation, we have found it useful to re-draw key elements of this figure, deriving relevant algebraic or trigonometric relations along the way. Because the "oblong" ellipse in Maclaurin's figure is oriented such that the axis of revolution — the symmetry axis — is horizontal, and because we prefer to associate the <math>~z</math> axis with the symmetry axis of the configuration, we will rather unconventionally refer to the horizontal axis as the <math>~z</math> (symmetry) axis. Adopting a cylindrical coordinate system furthermore means that, in our notation, the vertical axis will be the <math>~\varpi</math> axis.

	The blue ellipse centered on the origin of our plot represents Maclaurin's "ADBE" ellipse; in the expressions that follow, we will use <math>~a_1</math> and <math>~a_3</math> to refer, respectively, to the length of the semi-major axis (Maclaurin's line-segment "CA") and semi-minor axis (Maclaurin's line-segment "CD") of the ellipse; and, following Maclaurin, we will refer to the leftmost point on the ellipse (Maclaurin's point "A") as the "pole" which is positioned at coordinate <math>~(z, \varpi) = (-a_1, 0)</math>. (Note that in our plot, we have set <math>~a_1 = 1</math>, but otherwise we will leave its specification arbitrary.) In our plot, Maclaurin's "CNH" circle is represented by the green quarter circle centered on the pole. The blue ellipse is defined by the relation,		The blue ellipse centered on the origin of our plot represents Maclaurin's "ADBE" ellipse; in the expressions that follow, we will use <math>a_1</math> and <math>a_3</math> to refer, respectively, to the length of the semi-major axis (Maclaurin's line-segment "CA") and semi-minor axis (Maclaurin's line-segment "CD") of the ellipse; and, following Maclaurin, we will refer to the leftmost point on the ellipse (Maclaurin's point "A") as the "pole" which is positioned at coordinate <math>(z, \varpi) = (-a_1, 0)</math>. (Note that in our plot, we have set <math>a_1 = 1</math>, but otherwise we will leave its specification arbitrary.) In our plot, Maclaurin's "CNH" circle is represented by the green quarter circle centered on the pole. The blue ellipse is defined by the relation,
	~~<div align="center">~~
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\frac{z^2}{a_1^2} + \frac{\varpi^2}{a_3^2}</math>		<math>\frac{z^2}{a_1^2} + \frac{\varpi^2}{a_3^2}</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~1 </math>		<math>1 </math>
	</td>		</td>
	</tr>		</tr>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\Rightarrow ~~~~ \frac{z^2}{a_1^2} + (1-e^2)^{-1}\frac{\varpi^2}{a_1^2}</math>		<math>\Rightarrow ~~~~ \frac{z^2}{a_1^2} + (1-e^2)^{-1}\frac{\varpi^2}{a_1^2}</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~1 \, ,</math>		<math>1 \, ,</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	~~</div>~~
	and, because it is [[Appendix/Ramblings/ToroidalCoordinates#Off-center_Circle\|off-center]], the green quarter circle is defined by the relation,		and, because it is [[Appendix/Ramblings/ToroidalCoordinates#Off-center_Circle\|off-center]], the green quarter circle is defined by the relation,
	~~<div align="center">~~
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~(a_1 + z)^2 + \varpi^2</math>		<math>(a_1 + z)^2 + \varpi^2</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~a_1^2 </math>		<math>a_1^2 </math>
	</td>		</td>
	</tr>		</tr>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\Rightarrow ~~~~ \biggl(1 + \frac{z}{a_1} \biggr)^2 + \frac{\varpi^2}{a_1^2}</math>		<math>\Rightarrow ~~~~ \biggl(1 + \frac{z}{a_1} \biggr)^2 + \frac{\varpi^2}{a_1^2}</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~1 \, .</math>		<math>1 \, .</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	~~</div>~~

	The red line-segment in our plot is what Maclaurin refers to as "AM" which, in his words, is ''any right line from the pole A.'' We will use the variable, <math>~\alpha</math>, to identify the angle between this line-segment and the horizontal axis; in determining the gravitational attraction at pole A, Maclaurin ultimately sweeps this angle through the range, <math>~0 \le \alpha \le \tfrac{\pi}{2}</math>. Points lying along this line are defined by the relation,		The red line-segment in our plot is what Maclaurin refers to as "AM" which, in his words, is ''any right line from the pole A.'' We will use the variable, <math>\alpha</math>, to identify the angle between this line-segment and the horizontal axis; in determining the gravitational attraction at pole A, Maclaurin ultimately sweeps this angle through the range, <math>0 \le \alpha \le \tfrac{\pi}{2}</math>. Points lying along this line are defined by the relation,
	~~<div align="center">~~
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	</tr>		</tr>
	</table>		</table>
	~~</div>~~

	===Intersection With Circle===		===Intersection With Circle===

Joel2: /* Interpreting Maclaurin's Key Concluding Theorem */

2024-06-22T21:13:45Z

Interpreting Maclaurin's Key Concluding Theorem

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Joel2: Created page with "FORCETOC =Excerpts from A Treatise of Fluxions= {| class="MaclaurinSpheroids" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 150px; width: 150px; background-color:#ffeeee;" |Maclaurin's
Original Text
&
Analysis
(1742) |} Both Volume I and Volume II of Colin Maclaurin's "''A Tr..."

2024-06-22T18:03:29Z

Created page with "__FORCETOC__  =Excerpts from A Treatise of Fluxions= {| class="MaclaurinSpheroids" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 150px; width: 150px; background-color:#ffeeee;" |Maclaurin's Original Text & Analysis (1742) |} Both Volume I and Volume II of Colin Maclaurin's "''A Tr..."

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