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(Orthogonal Coordinates)
(Other Coordinate Pair in the Tangent Plane)
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   <td align="left">
   <td align="left">
<math>~
<math>~
-
(x - x_0) \frac{\partial \lambda_1}{\partial x} + (y - y_0)\frac{\partial \lambda_1}{\partial y} + (z - z_0)\frac{\partial \lambda_1}{\partial z}
+
(x - x_0) \biggl[ \frac{\partial \lambda_1}{\partial x} \biggr]_0
 +
+ (y - y_0) \biggl[\frac{\partial \lambda_1}{\partial y} \biggr]_0 
 +
+ (z - z_0) \biggl[\frac{\partial \lambda_1}{\partial z} \biggr]_0
</math>
</math>
   </td>
   </td>
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<math>~
<math>~
-
(x - x_0) \frac{\partial F}{\partial x} + (y - y_0)\frac{\partial F}{\partial y} + (z - z_0)\frac{\partial F}{\partial z}
+
(x - x_0) \biggl[ \frac{\partial F}{\partial x} \biggr]_0 
 +
+ (y - y_0) \biggl[\frac{\partial F}{\partial y} \biggr]_0 
 +
+ (z - z_0) \biggl[ \frac{\partial F}{\partial z} \biggr]_0
</math>
</math>
   </td>
   </td>
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<math>~
<math>~
-
(x - x_0) \biggl( \frac{x}{\lambda_1}\biggr) + (y - y_0)\biggl( \frac{q^2 y  }{ \lambda_1 } \biggr) + (z - z_0)\biggl( \frac{ p^2z }{ \lambda_1 } \biggr) \, .
+
(x - x_0) \biggl( \frac{x}{\lambda_1}\biggr)_0 + (y - y_0)\biggl( \frac{q^2 y  }{ \lambda_1 } \biggr)_0 + (z - z_0)\biggl( \frac{ p^2z }{ \lambda_1 } \biggr)_0
 +
</math>
 +
  </td>
 +
</tr>
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 +
<tr>
 +
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<math>~\Rightarrow~~~
 +
x \biggl( \frac{x}{\lambda_1}\biggr)_0 + y \biggl( \frac{q^2 y  }{ \lambda_1 } \biggr)_0 + z \biggl( \frac{ p^2z }{ \lambda_1 } \biggr)_0
 +
</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
x_0 \biggl( \frac{x}{\lambda_1}\biggr)_0 + y_0 \biggl( \frac{q^2 y  }{ \lambda_1 } \biggr)_0 + z_0 \biggl( \frac{ p^2z }{ \lambda_1 } \biggr)_0 \, .
</math>
</math>
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   </td>

Revision as of 20:03, 27 October 2020

Contents

Concentric Ellipsoidal (T6) 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 (T6) 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.

Orthogonal Coordinates

Primary (radial-like) Coordinate

We start by defining a "radial" coordinate whose values identify various concentric ellipsoidal shells,

~\lambda_1

~\equiv

~(x^2 + q^2 y^2 + p^2 z^2)^{1 / 2} \, .

When ~\lambda_1 = a, we obtain the standard definition of an ellipsoidal surface, it being understood that, ~q^2 = a^2/b^2 and ~p^2 = a^2/c^2. (We will assume that ~a > b > c, that is, ~p^2 > q^2 > 1.)

A vector, ~\bold{\hat{n}}, that is normal to the ~\lambda_1 = constant surface is given by the gradient of the function,

~F(x, y, z)

~\equiv

~(x^2 + q^2 y^2 + p^2 z^2)^{1 / 2} - \lambda_1 \, .

In Cartesian coordinates, this means,

~\bold{\hat{n}}(x, y, z)

~=

~
\hat\imath \biggl( \frac{\partial F}{\partial x} \biggr)
+ \hat\jmath \biggl( \frac{\partial F}{\partial y} \biggr)
+ \hat{k} \biggl( \frac{\partial F}{\partial z} \biggr)

 

~=

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

 

~=

~
\hat\imath \biggl( \frac{x}{\lambda_1} \biggr)
+ \hat\jmath \biggl( \frac{q^2y}{\lambda_1} \biggr)
+ \hat\jmath \biggl(\frac{p^2 z}{\lambda_1} \biggr) \, ,

where it is understood that this expression is only to be evaluated at points, ~(x, y, z), that lie on the selected ~\lambda_1 surface — that is, at points for which the function, ~F(x,y,z) = 0. The length of this normal vector is given by the expression,

~[ \bold{\hat{n}} \cdot \bold{\hat{n}} ]^{1 / 2}

~=

~
\biggl[ \biggl( \frac{\partial F}{\partial x} \biggr)^2 + \biggl( \frac{\partial F}{\partial y} \biggr)^2 + \biggl( \frac{\partial F}{\partial z} \biggr)^2 \biggr]^{1 / 2}

 

~=

~
\biggl[ \biggl( \frac{x}{\lambda_1} \biggr)^2
+ \biggl( \frac{q^2y}{\lambda_1} \biggr)^2
+ \biggl(\frac{p^2 z}{\lambda_1} \biggr)^2 \biggr]^{1 / 2}

 

~=

~
\frac{1}{\lambda_1 \ell_{3D}}

where,

~\ell_{3D}

~\equiv

~\biggl[ x^2 + q^4y^2 + p^4 z^2 \biggr]^{- 1 / 2} \, .

It is therefore clear that the properly normalized normal unit vector that should be associated with any ~\lambda_1 = constant ellipsoidal surface is,

~\hat{e}_1

~\equiv

~
\frac{ \bold\hat{n} }{ [ \bold{\hat{n}} \cdot \bold{\hat{n}} ]^{1 / 2} }
=
\hat\imath (x \ell_{3D}) + \hat\jmath (q^2y \ell_{3D}) + \hat\jmath (p^2 z \ell_{3D}) \, .

From our accompanying discussion of direction cosines, it is clear, as well, that the scale factor associated with the ~\lambda_1 coordinate is,

~h_1^2

~=

~\lambda_1^2 \ell_{3D}^2 \, .

We can also fill in the top line of our direction-cosines table, namely,


Direction Cosines for T6 Coordinates
~\gamma_{ni} = h_n \biggl( \frac{\partial \lambda_n}{\partial x_i}\biggr)

~n ~i = x, y, z
~1  

~x\ell_{3D}
 

~q^2 y \ell_{3D} ~p^2 z \ell_{3D}
~2

 
---
 

 
---
 

 
---
 

~3

 
---
 

 
---
 

 
---
 

Other Coordinate Pair in the Tangent Plane

Let's focus on a particular point on the ~\lambda_1 = constant surface, ~(x_0, y_0, z_0), that necessarily satisfies the function, ~F(x_0, y_0, z_0) = 0. We have already derived the expression for the unit vector that is normal to the ellipsoidal surface at this point, namely,

~\hat{e}_1

~\equiv

~
\hat\imath (x_0 \ell_{3D}) + \hat\jmath (q^2y_0 \ell_{3D}) + \hat\jmath (p^2 z_0 \ell_{3D}) \, ,

where, for this specific point on the surface,

~\ell_{3D}

~=

~\biggl[ x_0^2 + q^4y_0^2 + p^4 z_0^2 \biggr]^{- 1 / 2} \, .

The two-dimensional plane that is tangent to the ~\lambda_1 = constant surface at this point is given by the expression,

~0

~=

~
(x - x_0) \biggl[ \frac{\partial \lambda_1}{\partial x} \biggr]_0 
+ (y - y_0) \biggl[\frac{\partial \lambda_1}{\partial y} \biggr]_0  
+ (z - z_0) \biggl[\frac{\partial \lambda_1}{\partial z} \biggr]_0

 

~=

~
(x - x_0) \biggl[ \frac{\partial F}{\partial x} \biggr]_0  
+ (y - y_0) \biggl[\frac{\partial F}{\partial y} \biggr]_0  
+ (z - z_0) \biggl[ \frac{\partial F}{\partial z} \biggr]_0

 

~=

~
(x - x_0) \biggl( \frac{x}{\lambda_1}\biggr)_0 + (y - y_0)\biggl( \frac{q^2 y  }{ \lambda_1 } \biggr)_0 + (z - z_0)\biggl( \frac{ p^2z }{ \lambda_1 } \biggr)_0

~\Rightarrow~~~
x \biggl( \frac{x}{\lambda_1}\biggr)_0 + y \biggl( \frac{q^2 y  }{ \lambda_1 } \biggr)_0 + z \biggl( \frac{ p^2z }{ \lambda_1 } \biggr)_0

~=

~
x_0 \biggl( \frac{x}{\lambda_1}\biggr)_0 + y_0 \biggl( \frac{q^2 y  }{ \lambda_1 } \biggr)_0 + z_0 \biggl( \frac{ p^2z }{ \lambda_1 } \biggr)_0 \, .

See Also

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2020 by Joel E. Tohline
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