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<math>~b_{2D} = \frac{1}{q} \biggl(\lambda_1^2 - p^2z_0^2 \biggr)^{1 / 2} \, .</math>
<math>~b_{2D} = \frac{1}{q} \biggl(\lambda_1^2 - p^2z_0^2 \biggr)^{1 / 2} \, .</math>
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At each point along this elliptic curve, the line that is tangent to the curve has a slope that can be determined by simply differentiating the equation that describes the curve, that is,
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<table border="0" cellpadding="5" align="center">
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<math>~0</math>
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<math>~=</math>
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<math>~\frac{2x dx}{a_{2D}^2} + \frac{2y dy}{b_{2D}^2}</math>
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<math>~\Rightarrow~~~\frac{dy}{dx}</math>
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<math>~=</math>
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<math>~- \frac{2x}{a_{2D}^2} \cdot \frac{b_{2D}^2}{2y} = - \frac{x}{q^2y} \, .</math>
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<math>~\Rightarrow~~~\Delta y</math>
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<math>~=</math>
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<math>~- \biggl( \frac{x}{q^2y} \biggr)\Delta x \, .</math>
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The unit vector that lies tangent to any point on this elliptical curve will be described by the expression,
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<math>~\hat{e}_2</math>
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<math>~=</math>
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<math>~
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\hat\imath~ \biggl\{ \frac{\Delta x}{[ (\Delta x)^2 + (\Delta y)^2 ]^{1 / 2}} \biggr\}
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\hat\jmath~ \biggl\{ \frac{\Delta y}{[ (\Delta x)^2 + (\Delta y)^2 ]^{1 / 2}} \biggr\} </math>
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&nbsp;
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<math>~=</math>
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<math>~
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\hat\imath~ \biggl\{ \frac{1}{[ 1 + x^2/(q^4y^2) ]^{1 / 2}} \biggr\}
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-
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\hat\jmath~ \biggl\{ \frac{x/(q^2y)}{[ 1 + x^2/(q^4y^2) ]^{1 / 2}} \biggr\} </math>
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&nbsp;
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<math>~=</math>
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<math>~
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\hat\imath~ \biggl\{ \frac{q^2y}{[ x^2 + q^4y^2 ]^{1 / 2}} \biggr\}
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-
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\hat\jmath~ \biggl\{ \frac{x}{[ x^2 + q^4y^2  ]^{1 / 2}} \biggr\} \, .</math>
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As we have discovered, the coordinate that gives rise to this unit vector is,
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<math>~\lambda_2</math>
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<math>~=</math>
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<math>~\frac{x}{y^{1/q^2}} \, .</math>
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   </td>
</tr>
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Revision as of 23:24, 18 January 2021

Contents

Concentric Ellipsoidal (T8) Coordinates

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

Building on our general introduction to Direction Cosines in the context of orthogonal curvilinear coordinate systems, and on our previous development of T3 (concentric oblate-spheroidal) and T5 (concentric elliptic) coordinate systems, here we explore the creation of a concentric ellipsoidal (T8) coordinate system. This is motivated by our desire to construct a fully analytically prescribable model of a nonuniform-density ellipsoidal configuration that is an analog to Riemann S-Type ellipsoids.

Note that, in a separate but closely related discussion, we made attempts to define this coordinate system, numbering the trials up through "T7." In this "T7" effort, we were able to define a set of three, mutually orthogonal unit vectors that should work to define a fully three-dimensional, concentric ellipsoidal coordinate system. But we were unable to figure out what coordinate function, ~\lambda_3(x, y, z), was associated with the third unit vector. In addition, we found the ~\lambda_2 coordinate to be rather strange in that it was not oriented in a manner that resembled the classic spherical coordinate system. Here we begin by redefining the ~\lambda_2 coordinate such that its associated ~\hat{e}_3 unit vector lies parallel to the x-y plane.

Realigning the Second Coordinate

The first coordinate remains the same as before, namely,

~\lambda_1^2

~=

~x^2 + q^2 y^2 + p^2 z^2 \, .

This may be rewritten as,

~1

~=

~\biggl( \frac{x}{a}\biggr)^2 + \biggl( \frac{y}{b}\biggr)^2 + \biggl(\frac{z}{c}\biggr)^2 \, ,

where,

~a = \lambda_1 \, ,

      

~b = \frac{\lambda_1}{q} \, ,

      

~c = \frac{\lambda_1}{p} \, .

By specifying the value of ~z = z_0 < c, as well as the value of ~\lambda_1, we are picking a plane that lies parallel to, but a distance ~z_0 above, the equatorial plane. The elliptical curve that defines the intersection of the ~\lambda_1-constant surface with this plane is defined by the expression,

~\lambda_1^2 - p^2z_0^2

~=

~x^2 + q^2 y^2

~\Rightarrow~~~1

~=

~\biggl( \frac{x}{a_{2D}}\biggr)^2 + \biggl( \frac{y}{b_{2D}}\biggr)^2 \, ,

where,

~a_{2D} = \biggl(\lambda_1^2 - p^2z_0^2 \biggr)^{1 / 2} \, ,

      

~b_{2D} = \frac{1}{q} \biggl(\lambda_1^2 - p^2z_0^2 \biggr)^{1 / 2} \, .

At each point along this elliptic curve, the line that is tangent to the curve has a slope that can be determined by simply differentiating the equation that describes the curve, that is,

~0

~=

~\frac{2x dx}{a_{2D}^2} + \frac{2y dy}{b_{2D}^2}

~\Rightarrow~~~\frac{dy}{dx}

~=

~- \frac{2x}{a_{2D}^2} \cdot \frac{b_{2D}^2}{2y} = - \frac{x}{q^2y} \, .

~\Rightarrow~~~\Delta y

~=

~- \biggl( \frac{x}{q^2y} \biggr)\Delta x \, .

The unit vector that lies tangent to any point on this elliptical curve will be described by the expression,

~\hat{e}_2

~=

~ \hat\imath~ \biggl\{ \frac{\Delta x}{[ (\Delta x)^2 + (\Delta y)^2 ]^{1 / 2}} \biggr\} + \hat\jmath~ \biggl\{ \frac{\Delta y}{[ (\Delta x)^2 + (\Delta y)^2 ]^{1 / 2}} \biggr\}

 

~=

~ \hat\imath~ \biggl\{ \frac{1}{[ 1 + x^2/(q^4y^2) ]^{1 / 2}} \biggr\} - \hat\jmath~ \biggl\{ \frac{x/(q^2y)}{[ 1 + x^2/(q^4y^2) ]^{1 / 2}} \biggr\}

 

~=

~ \hat\imath~ \biggl\{ \frac{q^2y}{[ x^2 + q^4y^2 ]^{1 / 2}} \biggr\} - \hat\jmath~ \biggl\{ \frac{x}{[ x^2 + q^4y^2 ]^{1 / 2}} \biggr\} \, .

As we have discovered, the coordinate that gives rise to this unit vector is,

~\lambda_2

~=

~\frac{x}{y^{1/q^2}} \, .

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2020 by Joel E. Tohline
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