Difference between revisions of "User:Tohline/ThreeDimensionalConfigurations/ChallengesPt2"

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==Motivation==
==Motivation==
===Example Equilibrium Model===
These key parameters have been drawn from[[User:Tohline/Appendix/References#EFE|[<font color="red">EFE</font>] ]] Chapter 7, Table XIII (p. 170):
<table width="80%" align="center" cellpadding="8" border="0">
<tr><td align="left"><math>~a = a_1 = 1</math></td></tr>
<tr><td align="left"><math>~b = a_2 = 1.25</math></td></tr>
<tr><td align="left"><math>~c = a_3 = 0.4703</math></td></tr>
</table>
===COLLADA-Based Representation===


As has been described in an [[User:Tohline/ThreeDimensionalConfigurations/RiemannTypeI#Figure3|accompanying discussion of Riemann Type 1 ellipsoids]], we have used COLLADA to construct  an animated and interactive 3D scene that displays in purple the surface of an example Type I ellipsoid; panels a and b of Figure 1 show what this ellipsoid looks like when viewed from two different perspectives.  (As a reminder &#8212; see the [[#explanation| explanation accompanying Figure 2 of that accompanying discussion]] &#8212; the ellipsoid is tilted about the x-coordinate axis at an angle of 61.25&deg; to the equilibrium spin axis, which is shown in green.)  Yellow markers also have been placed in this 3D scene at each of the coordinate locations specified in the [[#ExampleTrajectories|table that accompanies that discussion]].  From the perspective presented in Figure 3b, we can immediately identify three separate, nearly circular trajectories; the largest one corresponds to our choice of z<sub>0</sub> = -0.25, the smallest corresponds to our choice of z<sub>0</sub> = -0.60, and the one of intermediate size correspond to our choice of z<sub>0</sub> = -0.4310.  When viewed from the perspective presented in Figure 1a, we see that these three trajectories define three separate planes; each plane is tipped at an angle of &theta; = -19.02&deg; to the ''untilted'' equatorial, x-y plane of the purple ellipsoid.
As has been described in an [[User:Tohline/ThreeDimensionalConfigurations/RiemannTypeI#Figure3|accompanying discussion of Riemann Type 1 ellipsoids]], we have used COLLADA to construct  an animated and interactive 3D scene that displays in purple the surface of an example Type I ellipsoid; panels a and b of Figure 1 show what this ellipsoid looks like when viewed from two different perspectives.  (As a reminder &#8212; see the [[#explanation| explanation accompanying Figure 2 of that accompanying discussion]] &#8212; the ellipsoid is tilted about the x-coordinate axis at an angle of 61.25&deg; to the equilibrium spin axis, which is shown in green.)  Yellow markers also have been placed in this 3D scene at each of the coordinate locations specified in the [[#ExampleTrajectories|table that accompanies that discussion]].  From the perspective presented in Figure 3b, we can immediately identify three separate, nearly circular trajectories; the largest one corresponds to our choice of z<sub>0</sub> = -0.25, the smallest corresponds to our choice of z<sub>0</sub> = -0.60, and the one of intermediate size correspond to our choice of z<sub>0</sub> = -0.4310.  When viewed from the perspective presented in Figure 1a, we see that these three trajectories define three separate planes; each plane is tipped at an angle of &theta; = -19.02&deg; to the ''untilted'' equatorial, x-y plane of the purple ellipsoid.

Revision as of 17:37, 7 April 2021

Challenges Constructing Ellipsoidal-Like Configurations (Pt. 2)

This chapter extends an accompanying chapter titled, Construction Challenges (Pt. 1). The focus here is on an SCF technique that will incorporate specification of a Lagrangian flow-flied.

Whitworth's (1981) Isothermal Free-Energy Surface
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Motivation

Example Equilibrium Model

These key parameters have been drawn from[EFE] Chapter 7, Table XIII (p. 170):

<math>~a = a_1 = 1</math>
<math>~b = a_2 = 1.25</math>
<math>~c = a_3 = 0.4703</math>

COLLADA-Based Representation

As has been described in an accompanying discussion of Riemann Type 1 ellipsoids, we have used COLLADA to construct an animated and interactive 3D scene that displays in purple the surface of an example Type I ellipsoid; panels a and b of Figure 1 show what this ellipsoid looks like when viewed from two different perspectives. (As a reminder — see the explanation accompanying Figure 2 of that accompanying discussion — the ellipsoid is tilted about the x-coordinate axis at an angle of 61.25° to the equilibrium spin axis, which is shown in green.) Yellow markers also have been placed in this 3D scene at each of the coordinate locations specified in the table that accompanies that discussion. From the perspective presented in Figure 3b, we can immediately identify three separate, nearly circular trajectories; the largest one corresponds to our choice of z0 = -0.25, the smallest corresponds to our choice of z0 = -0.60, and the one of intermediate size correspond to our choice of z0 = -0.4310. When viewed from the perspective presented in Figure 1a, we see that these three trajectories define three separate planes; each plane is tipped at an angle of θ = -19.02° to the untilted equatorial, x-y plane of the purple ellipsoid.

Figure 1a Figure 1b

EFE Model b41c385

EFE Model b41c385

file = Dropbox/3Dviewers/AutoRiemann/TypeI/Lagrange/TL15.lagrange.dae

Figure 1c

EFE Model b41c385

file = Dropbox/3Dviewers/RiemannModels/RiemannCalculations.xlsx --- worksheet = TypeI_1b

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

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