User:Tohline/Appendix/Ramblings/EllipticCylinderCoordinates
Elliptic Cylinder Coordinates
|
|---|
| | Tiled Menu | Tables of Content | Banner Video | Tohline Home Page | |
Background
Building on our general introduction to Direction Cosines in the context of orthogonal curvilinear coordinate systems, here we detail the properties of Elliptic Cylinder Coordinates. First, we will present this coordinate system in the manner described by [MF53]; second, we will provide an alternate presentation, obtained from Wikipedia; then, third, we will investigate whether or not a related coordinate system based on concentric (rather than confocal) elliptic surfaces can be satisfactorily described.
It is useful to keep in mind various properties of a set of confocal ellipses in which the location of the pair of foci is fixed at, <math>~(x, y) = (\pm~ c, 0)</math>, and the semi-major axis, <math>~a</math>, is the parameter. The relevant prescriptive relation is,
|
<math>~1</math> |
<math>~=</math> |
<math>~\frac{x^2}{a^2} + \frac{y^2}{a^2 - c^2}</math> for, <math>~a > c\, .</math> |
The semi-minor axis length, <math>~b</math>, and the eccentricity, <math>~e</math>, of the ellipse are, respectively,
|
<math>~b</math> |
<math>~=</math> |
<math>~(a^2 - c^2)^{1 / 2} \, ,</math> |
and, |
<math>~e\equiv \biggl[1 - \frac{b^2}{a^2} \biggr]^{1 / 2}</math> |
<math>~=</math> |
<math>~\frac{c}{a} \, .</math> |
The length, <math>~\ell_1</math>, of the chord that connects one focus to a point, <math>~P(x,y)</math>, on the ellipse is,
|
<math>~\ell_1</math> |
<math>~=</math> |
<math>~a + \biggl(\frac{c}{a}\biggr)x \, ;</math> |
and the length, <math>~\ell_2</math>, of the chord that connects the second focus to that same point on the ellipse is,
|
<math>~\ell_2</math> |
<math>~=</math> |
<math>~a - \biggl(\frac{c}{a}\biggr)x \, .</math> |
It is easy to see that, for any point on the ellipse, the sum of these two lengths is, <math>~2a</math>. It is worth noting as well that the associated <math>~y</math> coordinate of the relevant point can be obtained from the relation,
|
<math>~ y^2</math> |
<math>~=</math> |
<math>~\ell_1^2 - (c+x)^2</math> |
|
|
<math>~=</math> |
<math>~\ell_1^2 - (c+x)^2</math> |
MF53
Definition
From MF53's Table of Separable Coordinates in Three Dimensions (see their Chapter 5, beginning on p. 655), we find the following description of Elliptic Cylinder Coordinates (p. 657).
|
<math>~x</math> |
<math>~=</math> |
<math>~\xi_1 \xi_2 \, ;</math> |
|
<math>~y</math> |
<math>~=</math> |
<math>~\biggl[ (\xi_1^2 - d^2)(1 - \xi_2^2) \biggr]^{1 / 2} \, ;</math> |
|
<math>~z</math> |
<math>~=</math> |
<math>~\xi_3 \, .</math> |
Scale Factors
Appreciating that,
|
<math>~\frac{\partial y}{\partial \xi_1}</math> |
<math>~=</math> |
<math>~ +\biggl[ (\xi_1^2 - d^2)(1 - \xi_2^2) \biggr]^{- 1 / 2}\xi_1(1-\xi_2^2) \, , </math> and that, |
|
<math>~\frac{\partial y}{\partial \xi_2}</math> |
<math>~=</math> |
<math>~ - \biggl[ (\xi_1^2 - d^2)(1 - \xi_2^2) \biggr]^{- 1 / 2}\xi_2(\xi_1^2 - d^2) \, , </math> |
we find that the respective scale factors are given by the expressions,
|
<math>~ h_1^2</math> |
<math>~=</math> |
<math>~\biggl(\frac{\partial x}{\partial\xi_1} \biggr)^2 + \biggl(\frac{\partial y}{\partial\xi_1} \biggr)^2 + \biggl(\frac{\partial z}{\partial\xi_1} \biggr)^2 </math> |
|
|
<math>~=</math> |
<math>~\xi_2^2 +\biggl[ (\xi_1^2 - d^2)(1 - \xi_2^2) \biggr]^{- 1 }\xi_1^2 (1-\xi_2^2)^2 </math> |
|
|
<math>~=</math> |
<math>~ (\xi_1^2 - d^2)^{- 1 } [ (\xi_1^2 - d^2)\xi_2^2 +\xi_1^2 (1-\xi_2^2) ]</math> |
|
|
<math>~=</math> |
<math>~ \biggl[ \frac{ \xi_1^2 - d^2 \xi_2^2 }{\xi_1^2 - d^2} \biggr] \, ;</math> |
|
<math>~ h_2^2</math> |
<math>~=</math> |
<math>~\biggl(\frac{\partial x}{\partial\xi_2} \biggr)^2 + \biggl(\frac{\partial y}{\partial\xi_2} \biggr)^2 + \biggl(\frac{\partial z}{\partial\xi_2} \biggr)^2 </math> |
|
|
<math>~=</math> |
<math>~\xi_1^2 + \biggl[ (\xi_1^2 - d^2)(1 - \xi_2^2) \biggr]^{- 1 }\xi_2^2(\xi_1^2 - d^2)^2 </math> |
|
|
<math>~=</math> |
<math>~(1 - \xi_2^2)^{- 1 } [\xi_1^2(1 - \xi_2^2) + \xi_2^2(\xi_1^2 - d^2) ]</math> |
|
|
<math>~=</math> |
<math>~\biggl[ \frac{ \xi_1^2 - d^2 \xi_2^2 }{1 - \xi_2^2} \biggr] \, ;</math> |
|
<math>~ h_3^2</math> |
<math>~=</math> |
<math>~\biggl(\frac{\partial x}{\partial\xi_3} \biggr)^2 + \biggl(\frac{\partial y}{\partial\xi_3} \biggr)^2 + \biggl(\frac{\partial z}{\partial\xi_3} \biggr)^2 </math> |
|
|
<math>~=</math> |
<math>~1 \, . </math> |
These match the scale-factor expressions found in MF53.
Inverting Coordinate Mapping
Inverting the original coordinate mappings, we find,
|
<math>~y^2</math> |
<math>~=</math> |
<math>~(\xi_1^2 - d^2)\biggl[ 1 - \biggl(\frac{x}{\xi_1}\biggr)^2 \biggr] </math> |
|
<math>~\Rightarrow ~~~0</math> |
<math>~=</math> |
<math>~(\xi_1^2 - d^2) ( \xi_1^2 - x^2 ) - \xi_1^2 y^2</math> |
|
|
<math>~=</math> |
<math>~(\xi_1^2 - d^2) \xi_1^2 - (\xi_1^2 - d^2) x^2 - \xi_1^2 y^2</math> |
|
|
<math>~=</math> |
<math>~ \xi_1^4 - \xi_1^2 (d^2 + x^2 + y^2) + d^2 x^2 </math> |
|
<math>~\Rightarrow~~~ \xi_1^2</math> |
<math>~=</math> |
<math>~ \frac{1}{2}\biggl\{ -(d^2 + x^2 + y^2) \pm \biggl[ (d^2 + x^2 + y^2)^2 + 4d^2 x^2 \biggr]^{1 / 2} \biggr\} </math> |
Only the superior — that is, only the positive — sign will ensure positive values of <math>~\xi_1^2</math>, so in summary we have,
| ||||||||||||
See Also
|
|
|---|
|
© 2014 - 2021 by Joel E. Tohline |