Jet53man at 00:43, 12 December 2021

2021-12-12T00:43:45Z

← Older revision		Revision as of 20:43, 11 December 2021
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	=Another S-type Example b74c692=		=Another S-type Example b74c692=

	This chapter is an extension of the chapter we have titled, "[[~~User:Tohline/~~ThreeDimensionalConfigurations/MeetsCOLLADAandOculusRiftS\|Riemann Meets COLLADA & Oculus Rift S]]." In that chapter we used as our first example of a Riemann S-type ellipsoid the model with parameters, <b>(b/a, c/a) = (0.41, 0.385)</b>. Here we construct a model with parameters, <b>(b/a, c/a) = (0.74, 0.692)</b>. Other closely related chapters are listed below under the heading, "[[#See_Also\|See Also]]".		This chapter is an extension of the chapter we have titled, "[[ThreeDimensionalConfigurations/MeetsCOLLADAandOculusRiftS\|Riemann Meets COLLADA & Oculus Rift S]]." In that chapter we used as our first example of a Riemann S-type ellipsoid the model with parameters, <b>(b/a, c/a) = (0.41, 0.385)</b>. Here we construct a model with parameters, <b>(b/a, c/a) = (0.74, 0.692)</b>. Other closely related chapters are listed below under the heading, "[[#See_Also\|See Also]]".

	==Key Physical Parameters==		==Key Physical Parameters==

	The model that we have chosen to use in our second successful construction of a COLLADA-based, 3D and interactive animation has the following properties; this model has been selected from [[~~User:Tohline/~~ThreeDimensionalConfigurations/RiemannStype#Table2\|Table 2 of our accompanying discussion of Riemann S-type ellipsoids]]:		The model that we have chosen to use in our second successful construction of a COLLADA-based, 3D and interactive animation has the following properties; this model has been selected from [[ThreeDimensionalConfigurations/RiemannStype#Table2\|Table 2 of our accompanying discussion of Riemann S-type ellipsoids]]:

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

	The subscript "EFE" on Ω and λ means that the relevant frequency is given in units that have been adopted in [<b>[[~~User:Tohline/~~Appendix/References#EFE\|<font color="red">EFE</font>]]</b>], that is, in units of <math>~[\pi G\rho]^{1 / 2}</math>. In Figure 1a, the yellow circular marker, that has been placed where the pair of purple dashed lines cross, identifies the location of this model in the "c/a versus b/a" ''[[~~User:Tohline/~~ThreeDimensionalConfigurations/RiemannStype#Fig2\|EFE Diagram]]'' that appears as Figure 2 on p. 902 of [http://adsabs.harvard.edu/abs/1965ApJ...142..890C S. Chandrasekhar (1965)]; essentially the same diagram appears in §49 (p. 147) of [<b>[[~~User:Tohline/~~Appendix/References#EFE\|<font color="red">EFE</font>]]</b>].		The subscript "EFE" on Ω and λ means that the relevant frequency is given in units that have been adopted in [<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>], that is, in units of <math>~[\pi G\rho]^{1 / 2}</math>. In Figure 1a, the yellow circular marker, that has been placed where the pair of purple dashed lines cross, identifies the location of this model in the "c/a versus b/a" ''[[ThreeDimensionalConfigurations/RiemannStype#Fig2\|EFE Diagram]]'' that appears as Figure 2 on p. 902 of [http://adsabs.harvard.edu/abs/1965ApJ...142..890C S. Chandrasekhar (1965)]; essentially the same diagram appears in §49 (p. 147) of [<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>].

	==Coding Steps==		==Coding Steps==
	Here we begin with a working model of [[~~User:Tohline/~~Appendix/Ramblings/RiemannMeetsOculus#S-Type_Ellipsoid_b41c385\|b41c385]] and use incremental changes in the COLLADA-based code to construct a working model of b74c692.		Here we begin with a working model of [[Appendix/Ramblings/RiemannMeetsOculus#S-Type_Ellipsoid_b41c385\|b41c385]] and use incremental changes in the COLLADA-based code to construct a working model of b74c692.

	<ul>		<ul>
	<li>Pulling from the earlier modeling subsection titled, [[~~User:Tohline/~~Appendix/Ramblings/RiemannMeetsOculus#Final_Touches\|Final Touches]] …		<li>Pulling from the earlier modeling subsection titled, [[Appendix/Ramblings/RiemannMeetsOculus#Final_Touches\|Final Touches]] …
	<ul>		<ul>
	<li>		<li>
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	</li>		</li>

	<li>Pulling from a different earlier subsection titled, [[~~User:Tohline/~~Appendix/Ramblings/RiemannMeetsOculus#Multiple_Lagrangian_Fluid_Elements\|Multiple Lagrangian Fluid Elements]] …		<li>Pulling from a different earlier subsection titled, [[Appendix/Ramblings/RiemannMeetsOculus#Multiple_Lagrangian_Fluid_Elements\|Multiple Lagrangian Fluid Elements]] …
	<ul>		<ul>
	<li>		<li>
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	===location_X===		===location_X===
	Here we illustrate the motion of one Lagrangian fluid element as it moves along an elliptical path (b/a aspect ratio) in the equatorial plane of the Riemann ellipsoid. First we acknowledge that every fluid element completes one full orbit in a COLLADA-based time of, TIME<sub>λ</sub> = 4/\|λ<sub>EFE</sub>\|. Dividing the orbit into 50 equally spaced intervals of time therefore specifies a time increment, ΔT<sub>λ</sub> = 4/(50 \|λ<sub>EFE</sub>\|). Drawing from a [[~~User:Tohline/~~ThreeDimensionalConfigurations/RiemannStype#Feeding_a_3D_Animation\|related accompanying discussion]], we recognize that at each of these points in time — that is, for n = 1 → 50 — the x and y coordinate locations of the fluid element are given by the expressions,		Here we illustrate the motion of one Lagrangian fluid element as it moves along an elliptical path (b/a aspect ratio) in the equatorial plane of the Riemann ellipsoid. First we acknowledge that every fluid element completes one full orbit in a COLLADA-based time of, TIME<sub>λ</sub> = 4/\|λ<sub>EFE</sub>\|. Dividing the orbit into 50 equally spaced intervals of time therefore specifies a time increment, ΔT<sub>λ</sub> = 4/(50 \|λ<sub>EFE</sub>\|). Drawing from a [[ThreeDimensionalConfigurations/RiemannStype#Feeding_a_3D_Animation\|related accompanying discussion]], we recognize that at each of these points in time — that is, for n = 1 → 50 — the x and y coordinate locations of the fluid element are given by the expressions,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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

	As we have [[~~User:Tohline/~~Appendix/Ramblings/RiemannMeetsOculus#Final_Touches\|detailed in a parallel discussion]], at each discrete time step, the equivalent '''<matrix>''' instruction should be of the form, <table border="0" align="center" cellpadding="8"><tr><td align="center"><matrix>[ 0 ]   [ 0 ]   [ -S<sub>x</sub> ]   [ T<sub>x</sub> ]     [ S<sub>y</sub> · sin(γ) ]   [ S<sub>y</sub> · cos(γ) ]   [ 0 ]   [ T<sub>y</sub> ]     [ S<sub>z</sub> · cos(γ) ]   [ -S<sub>z</sub> · sin(γ) ]   [ 0]   [ T<sub>z</sub> ]     [ 0 ]   [ 0 ]   [ 0 ]   [ 1 ]</matrix> .</td></tr></table>		As we have [[Appendix/Ramblings/RiemannMeetsOculus#Final_Touches\|detailed in a parallel discussion]], at each discrete time step, the equivalent '''<matrix>''' instruction should be of the form, <table border="0" align="center" cellpadding="8"><tr><td align="center"><matrix>[ 0 ]   [ 0 ]   [ -S<sub>x</sub> ]   [ T<sub>x</sub> ]     [ S<sub>y</sub> · sin(γ) ]   [ S<sub>y</sub> · cos(γ) ]   [ 0 ]   [ T<sub>y</sub> ]     [ S<sub>z</sub> · cos(γ) ]   [ -S<sub>z</sub> · sin(γ) ]   [ 0]   [ T<sub>z</sub> ]     [ 0 ]   [ 0 ]   [ 0 ]   [ 1 ]</matrix> .</td></tr></table>

	Note that after the clock was originally built as a separate object for these visual scenes, I realized that I needed the starting angle for both hands to be γ = +270° = -90° in order for them to be properly registered at 0<sup>h</sup>0<sup>m</sup> at the start of each animation. This is why the table of signs and cosines (immediately above) starts at -90°.		Note that after the clock was originally built as a separate object for these visual scenes, I realized that I needed the starting angle for both hands to be γ = +270° = -90° in order for them to be properly registered at 0<sup>h</sup>0<sup>m</sup> at the start of each animation. This is why the table of signs and cosines (immediately above) starts at -90°.

Jet53man: /* See Also */

2021-12-12T00:37:00Z

Jet53man: Created page with "FORCETOC =Another S-type Example b74c692= This chapter is an extension of the chapter we have titled, "User:Tohline/ThreeDimension..."

2021-07-24T21:29:37Z

Created page with "__FORCETOC__  =Another S-type Example b74c692= This chapter is an extension of the chapter we have titled, "User:Tohline/ThreeDimension..."

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← Older revision		Revision as of 20:37, 11 December 2021
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	=See Also=		=See Also=

	* Discussion of [[~~User:Tohline/~~ThreeDimensionalConfigurations/RiemannStype\|Ou's Riemann-Like Ellipsoids]]		* Discussion of [[ThreeDimensionalConfigurations/RiemannStype\|Ou's Riemann-Like Ellipsoids]]

	* [[~~User:Tohline/~~ThreeDimensionalConfigurations/MeetsCOLLADAandOculusRiftS\|Riemann Meets COLLADA & Oculus Rift S]]: Example <b>(b/a, c/a) = (0.41, 0.385)</b>		* [[ThreeDimensionalConfigurations/MeetsCOLLADAandOculusRiftS\|Riemann Meets COLLADA & Oculus Rift S]]: Example <b>(b/a, c/a) = (0.41, 0.385)</b>
	** [[~~User:Tohline/~~Appendix/Ramblings/VirtualReality#Virtual_Reality_and_3D_Printing\|Virtual Reality and 3D Printing]]		** [[Appendix/Ramblings/VirtualReality#Virtual_Reality_and_3D_Printing\|Virtual Reality and 3D Printing]]
	** [[~~User:Tohline/~~Appendix/Ramblings/~~OculusRift_S~~\|Success Importing Animated Scene into Oculus Rift S]]		** [[Appendix/Ramblings/OculusRiftS\|Success Importing Animated Scene into Oculus Rift S]]
	** [[~~User:Tohline/~~Appendix/Ramblings/RiemannMeetsOculus\|Carefully (Re)Build Riemann Type S Ellipsoids Inside Oculus Rift Environment]]		** [[Appendix/Ramblings/RiemannMeetsOculus\|Carefully (Re)Build Riemann Type S Ellipsoids Inside Oculus Rift Environment]]
	** Other Example S-type Riemann Ellipsoids:		** Other Example S-type Riemann Ellipsoids:
	*** <b>[[~~User:Tohline/~~Appendix/Ramblings/RiemannB90C333\|(b/a, c/a) = (0.90, 0.333)]]</b>		*** <b>[[Appendix/Ramblings/RiemannB90C333\|(b/a, c/a) = (0.90, 0.333)]]</b>
	*** <b>[[~~User:Tohline/~~Appendix/Ramblings/RiemannB74C692\|(b/a, c/a) = (0.74, 0.692)]]</b>		*** <b>[[Appendix/Ramblings/RiemannB74C692\|(b/a, c/a) = (0.74, 0.692)]]</b>
	*** <b>[[~~User:Tohline/~~Appendix/Ramblings/RiemannB28C256\|(b/a, c/a) = (0.28, 0.256)]]</b>		*** <b>[[Appendix/Ramblings/RiemannB28C256\|(b/a, c/a) = (0.28, 0.256)]]</b>

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Appendix/Ramblings/RiemannB74C692 - Revision history

Jet53man at 00:43, 12 December 2021

Jet53man: /* See Also */

Jet53man: Created page with "__FORCETOC__ =Another S-type Example b74c692= This chapter is an extension of the chapter we have titled, "User:Tohline/ThreeDimension..."

Jet53man: Created page with "FORCETOC =Another S-type Example b74c692= This chapter is an extension of the chapter we have titled, "User:Tohline/ThreeDimension..."