Appendix/Ramblings/RiemannB28C256

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Another S-type Example b28c256

This chapter is an extension of the chapter we have titled, "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/a, c/a) = (0.41, 0.385). Here we construct a model with parameters, (b/a, c/a) = (0.28, 0.256). Other closely related chapters are listed below under the heading, "See Also".

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 Table 2 of our accompanying discussion of Riemann S-type ellipsoids:

Figure 1a
 
EFE Parameter Space

 

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\frac{b}{a} = 0.28}

 

Figure 1b
EFE Model b28c256

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\frac{c}{a} = 0.256}

Direct

 

Adjoint

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\Omega_\mathrm{EFE} = 0.456676}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\Omega_\mathrm{EFE} = - 0.020692}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\lambda_\mathrm{EFE} = 0.020692}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\lambda_\mathrm{EFE} = - 0.456676}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~f = - 0.174510}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~f = - 85.0007}

The subscript "EFE" on Ω and λ means that the relevant frequency is given in units that have been adopted in [EFE], that is, in units of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~[\pi G\rho]^{1 / 2}} . 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" EFE Diagram that appears as Figure 2 on p. 902 of S. Chandrasekhar (1965); essentially the same diagram appears in §49 (p. 147) of [EFE].

Coding Steps

Part A

Here we begin with a working model of b90c333 and use incremental changes in the COLLADA-based code to construct a working model of b28c256.

  • This pair of starting models has been copied from the successful modeling of Riemann S-type ellipsoids that have (b/a, c/a) = (0.90, 0.333)
    • Inertial Frame: [KEEP] FastInertial80.dae [04 June 2020]
    • Rotating Frame: [KEEP] FastRot79.dae [04 June 2020]
  • Pencil90.dae
    Identical to FastRot79.dae except wall-mounted labeling has been changed to reflect new values of b/a and c/a. This works in both visualization venues.
  • Pencil91.dae
    Inserted correct surface geometry of this rapidly rotating Riemann ellipsoid. This works in both visualization venues.
  • Pencil92.dae
    Inserted the correct animated depiction of the 9 Lagrangian fluid elements.
  • Pencil93.dae
    Placed 51 yellow, equatorial-plane markers. This works in both visualization venues
  • Pencil94.dae
    Finished specifying correct behavior of clock, which gives the final, fully functional model. This works in both visualization venues
  • PencilInertial95.dae
    Flipped from rotating- to inertial-frame of reference. This works in both visualization venues
  • PencilInertial96.dae
    Enlarged red "Lagrange01" marker from 0.03 to 0.075; and changed transparency of ellipsoid surface to (totally opaque) 1.0. This works in both visualization venues


Best b28c256 Models

The example models created for display in the Oculus Rift S are the following:

  1. Inertial Frame: [KEEP] PencilInertial96.dae [04 June 2020]
  2. Rotating Frame: [KEEP] Pencil94.dae [04 June 2020]

COLLADA Model Files

Direct Configurations

b28c256DI.dae [Direct Inertial Frame]    …     a COLLADA code containing nnnn lines of <xml>-formatted ASCII text
Original filename used above: xxx.dae

<?xml version="1.0" encoding="UTF-8" standalone="no" ?>


b28c256DRot.dae [Direct Rotating Frame]    …     a COLLADA code containing nnnn lines of <xml>-formatted ASCII text
Original filename used above: xxx.dae

<?xml version="1.0" encoding="UTF-8" standalone="no" ?>

See Also


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