User:Tohline/Apps/MaclaurinSpheroids/GoogleBooks - Revision history

Tohline: /* Prolate Spheroid */ Insert link to additional discussion of "off-center circles" found when discussing toroidal coordinates

2015-09-21T22:58:07Z

Prolate Spheroid: Insert link to additional discussion of "off-center circles" found when discussing toroidal coordinates

← Older revision		Revision as of 22:58, 21 September 2015
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	and the green quarter circle is defined by the relation,		and, because it is [[User:Tohline/Appendix/Ramblings/ToroidalCoordinates#Off-center_Circle\|off-center]], the green quarter circle is defined by the relation,
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Tohline: /* Prolate Spheroid */ Adjust figure caption and text to better correspond with newly animated figure

2015-09-14T20:25:07Z

Prolate Spheroid: Adjust figure caption and text to better correspond with newly animated figure

← Older revision		Revision as of 20:25, 14 September 2015
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	In this ~~plot~~, the ellipse has an eccentricity, <math>~e = 0.55</math>, and the red line-segment ~~has been~~ drawn at an angle of <math>~\alpha ~~= 3~~\pi/8</math>.		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.
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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 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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	The red line-segment in our plot is what Maclaurin refers to as "AM," which 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,
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	The "inferior" root clearly identifies the intersection of the red line-segment with the pole of the prolate spheroid. The "superior" root identifies the point at which the red line-segment intersects and crosses the blue ellipse, ~~below~~ the pole.		The "inferior" root clearly identifies the intersection of the red line-segment with the pole of the prolate spheroid. The "superior" root identifies the point at which the red line-segment intersects and crosses the blue ellipse, away from the pole.

	===Generate Curve AKG===		===Generate Curve AKG===

Tohline: /* Prolate Spheroid */ Inserted Excel-produced animation of Maclaurin's Fig. 291, No. 2

2015-09-14T19:44:45Z

Prolate Spheroid: Inserted Excel-produced animation of Maclaurin's Fig. 291, No. 2

← Older revision		Revision as of 19:44, 14 September 2015
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	[[File:MyFig291Pt2C.png\|380px\|center\|Our Construction]]		<!-- [[File:MyFig291Pt2C.png\|380px\|center\|Our Construction]] -->
			[[File:Maclaurin01.gif\|380px\|center\|Our Construction]]
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Tohline: /* Comparison With More Traditional Derivation */ Insert final "to be done" sentence

2015-09-10T14:51:31Z

Comparison With More Traditional Derivation: Insert final "to be done" sentence

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	This does not appear to match the other integrand. I must be misinterpreting something!		This does not appear to match the other integrand. I must be misinterpreting something!

			To be done: I suspect that Maclaurin's approach will look more familiar if I redo the "traditional" derivation from the point of view of a spherical coordinate system (positioning pole A at the origin), in which an axisymmetric volume element takes the form,
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			<math>~dV = 2\pi r^2 \sin\alpha dr d\alpha = 2\pi r^2 dr du_\mathrm{RN} \, ,</math>
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			where, from above, <math>u_\mathrm{RN} \equiv \cos\alpha \, .</math>

	=Related Discussions=		=Related Discussions=

Tohline: /* Excerpts from A Treatise of Fluxions */

2015-09-08T18:55:24Z

Excerpts from A Treatise of Fluxions

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	<th align="center" colspan="2">		<th align="center" colspan="2">
	Title Pages with Google-Books Links		Title Pages with Google-Books Links
	~~</th>~~
	~~<th align="center" colspan="2">~~
	~~Example Digitized Figure Plates~~
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	[[File:TitlePage2ndEdition01.png\|center\|border\|200px\|Volume II (2<sup>nd</sup> Ed., 1801)]]		[[File:TitlePage2ndEdition01.png\|center\|border\|200px\|Volume II (2<sup>nd</sup> Ed., 1801)]]
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			[https://books.google.com/books?id=HqZeAAAAcAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false Volume I (1<sup>st</sup> Ed., 1742)]
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			[https://books.google.com/books?id=xfQ7AQAAMAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false Volume II (2<sup>nd</sup> Ed., 1801)]
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			As is illustrated by the images displayed in the following two panels (click in one of the panels in order to view a higher-resolution image), each published "Figure Plate" was digitized (by Google Books) in segments that must be digitally pieced together in order to reconstruct some of the individual figures. See, for example, the discussion that references [[#Paragraph630\|Fig. 282, below]].


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			Example Digitized Figure Plates
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	<td align="center>		<td align="center>
	[[File:PlateXXXII_vol2.png\|center\|border\|250px\|Plate XXXII from Volume II (2<sup>nd</sup> Ed., 1801)]]		[[File:PlateXXXII_vol2.png\|center\|border\|250px\|Plate XXXII from Volume II (2<sup>nd</sup> Ed., 1801)]]
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	~~<td align="center">~~
	~~[https://books.google.com/books?id=HqZeAAAAcAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false Volume I (1<sup>st</sup> Ed., 1742)]~~
	~~</td>~~
	~~<td align="center">~~
	~~[https://books.google.com/books?id=xfQ7AQAAMAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false Volume II (2<sup>nd</sup> Ed., 1801)]~~
	~~</td>~~
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	Plate XXXII		Plate XXXII
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	As is illustrated in the two righthand panels of the above image quartet (click in one of the panels in order to view a higher-resolution image), each published "Figure Plate" was digitized (by Google Books) in segments that must be digitally pieced together in order to reconstruct some of the individual figures. See, for example, the discussion that references [[#Paragraph630\|Fig. 282, below]].

	<div style="clear:both"></div>		<div style="clear:both"></div>

Tohline: Insert "work in progress" notice

2015-09-08T18:48:20Z

Insert "work in progress" notice

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	This is precisely the expression for the coefficient, <math>~A_1</math>, that is [[User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids#Prolate_Spheroids_.28a1_.3E_a2_.3D_a3.29\|traditionally used in the expression for the gravitational potential inside homogeneous, prolate spheroids]].		This is precisely the expression for the coefficient, <math>~A_1</math>, that is [[User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids#Prolate_Spheroids_.28a1_.3E_a2_.3D_a3.29\|traditionally used in the expression for the gravitational potential inside homogeneous, prolate spheroids]].


			{{LSU_WorkInProgress}}
	===Comparison With More Traditional Derivation===		===Comparison With More Traditional Derivation===

Tohline: /* Comparison With More Traditional Derivation */

2015-09-07T23:12:15Z

Comparison With More Traditional Derivation

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	~~<math>~\biggl[\frac{\mathrm{AQ}}{a_1}\biggr] du_\mathrm{RN}</math>~~
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	<math>~-\biggl[\frac{(1-e^2)}{(2-e^2 -e^2\zeta)} \biggr]\biggl[\frac{(1-\zeta)^{1/2} }{ (2-e^2 -e^2\zeta)^{1/2}}\biggr] d\zeta \, .</math>		<math>~-\biggl[\frac{(1-e^2)}{(2-e^2 -e^2\zeta)} \biggr]\biggl[\frac{(1-\zeta)^{1/2} }{ (2-e^2 -e^2\zeta)^{1/2}}\biggr] d\zeta \, .</math>
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			<math>~=</math>
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			<math>~-\biggl[\frac{(1-e^2)}{(2-e^2 -e^2\zeta)} \biggr]\biggl[1 + \mathcal{Z}\biggr] d\zeta \, .</math>
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Tohline: /* Comparison With More Traditional Derivation */

2015-09-07T23:05:26Z

Comparison With More Traditional Derivation

← Older revision		Revision as of 23:05, 7 September 2015
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	<math>~\frac{(1-\zeta)}{ [ (2-e^2) - 2\zeta + e^2\zeta^2 ]^{1/2}} -1 \, .</math>		<math>~\frac{(1-\zeta)}{ [ (2-e^2) - 2\zeta + e^2\zeta^2 ]^{1/2}} -1 </math>
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			<math>~=</math>
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			<math>~\frac{(1-\zeta)^{1/2}}{ [ 2-e^2 - e^2\zeta ]^{1/2}} -1 \, .</math>
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	<math>~-\biggl[\frac{(1-e^2)(1-\zeta)^{1/2} }{ (2-e^2 -e^2\zeta)^{3/2}}\biggr] d\zeta \, .</math>		<math>~-\biggl[\frac{(1-e^2)(1-\zeta)^{1/2} }{ (2-e^2 -e^2\zeta)^{3/2}}\biggr] d\zeta </math>
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			<math>~\biggl[\frac{\mathrm{AQ}}{a_1}\biggr] du_\mathrm{RN}</math>
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			<math>~=</math>
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			<math>~-\biggl[\frac{(1-e^2)}{(2-e^2 -e^2\zeta)} \biggr]\biggl[\frac{(1-\zeta)^{1/2} }{ (2-e^2 -e^2\zeta)^{1/2}}\biggr] d\zeta \, .</math>
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Tohline: /* Prolate Spheroid */

2015-09-07T21:56:12Z

Prolate Spheroid

Show changes

Tohline: /* Prolate Spheroid */ Insert revised figure showing curve AKG

2015-09-06T21:03:46Z

Prolate Spheroid: Insert revised figure showing curve AKG

← Older revision		Revision as of 21:03, 6 September 2015
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	[[File:Figure291Pt2.png\|300px\|center\|Maclaurin (1742)]]		[[File:Figure291Pt2.png\|300px\|center\|Maclaurin (1742)]]
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	<td align="center~~" width="50%~~">		<td align="center">
	[[File:~~MyFig291Pt2A~~.png\|~~300px~~\|center\|Our Construction]]		[[File:MyFig291Pt2C.png\|380px\|center\|Our Construction]]
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	In this plot, the ellipse has an eccentricity, <math>~e = 0.55</math>, and the red line-segment has been drawn at an angle of <math>~\alpha = ~~0.7 (\tfrac{~~\pi~~}{2})~~</math>.		In this plot, the ellipse has an eccentricity, <math>~e = 0.55</math>, and the red line-segment has been drawn at an angle of <math>~\alpha = 3\pi/8</math>.
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