# Elliptic Cylinder Coordinates

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.

## MF53

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).

 $~x$ $~=$ $~\xi_1 \xi_2 \, ;$ $~y$ $~=$ $~\biggl[ (\xi_1^2 - d^2)(1 - \xi_2^2) \biggr]^{1 / 2} \, ;$ $~z$ $~=$ $~\xi_3 \, .$

Appreciating that,

 $~\frac{\partial y}{\partial \xi_1}$ $~=$ $~ +\biggl[ (\xi_1^2 - d^2)(1 - \xi_2^2) \biggr]^{- 1 / 2}\xi_1(1-\xi_2^2) \, ,$       and that, $~\frac{\partial y}{\partial \xi_2}$ $~=$ $~ - \biggl[ (\xi_1^2 - d^2)(1 - \xi_2^2) \biggr]^{- 1 / 2}\xi_2(\xi_1^2 - d^2) \, ,$

we find that the respective scale factors are given by the expressions,

 $~ h_1^2$ $~=$ $~\biggl(\frac{\partial x}{\partial\xi_1} \biggr)^2 + \biggl(\frac{\partial y}{\partial\xi_1} \biggr)^2 + \biggl(\frac{\partial z}{\partial\xi_1} \biggr)^2$ $~=$ $~\xi_2^2 +\biggl[ (\xi_1^2 - d^2)(1 - \xi_2^2) \biggr]^{- 1 }\xi_1^2 (1-\xi_2^2)^2$ $~=$ $~ (\xi_1^2 - d^2)^{- 1 } [ (\xi_1^2 - d^2)\xi_2^2 +\xi_1^2 (1-\xi_2^2) ]$ $~=$ $~ \biggl[ \frac{ \xi_1^2 - d^2 \xi_2^2 }{\xi_1^2 - d^2} \biggr] \, ;$ $~ h_2^2$ $~=$ $~\biggl(\frac{\partial x}{\partial\xi_2} \biggr)^2 + \biggl(\frac{\partial y}{\partial\xi_2} \biggr)^2 + \biggl(\frac{\partial z}{\partial\xi_2} \biggr)^2$ $~=$ $~\xi_1^2 + \biggl[ (\xi_1^2 - d^2)(1 - \xi_2^2) \biggr]^{- 1 }\xi_2^2(\xi_1^2 - d^2)^2$ $~=$ $~(1 - \xi_2^2)^{- 1 } [\xi_1^2(1 - \xi_2^2) + \xi_2^2(\xi_1^2 - d^2) ]$ $~=$ $~\biggl[ \frac{ \xi_1^2 - d^2 \xi_2^2 }{1 - \xi_2^2} \biggr] \, ;$ $~ h_3^2$ $~=$ $~\biggl(\frac{\partial x}{\partial\xi_3} \biggr)^2 + \biggl(\frac{\partial y}{\partial\xi_3} \biggr)^2 + \biggl(\frac{\partial z}{\partial\xi_3} \biggr)^2$ $~=$ $~1 \, .$

These match the scale-factor expressions found in MF53. Inverting the original coordinate mappings, we find,

 $~y^2$ $~=$ $~(\xi_1^2 - d^2)\biggl[ 1 - \biggl(\frac{x}{\xi_1}\biggr)^2 \biggr]$ $~\Rightarrow ~~~0$ $~=$ $~(\xi_1^2 - d^2) ( \xi_1^2 - x^2 ) - \xi_1^2 y^2$ $~=$ $~(\xi_1^2 - d^2) \xi_1^2 - (\xi_1^2 - d^2) x^2 - \xi_1^2 y^2$ $~=$ $~ \xi_1^4 - \xi_1^2 (d^2 + x^2 + y^2) + d^2 x^2$ $~\Rightarrow~~~ \xi_1^2$ $~=$ $~ \frac{1}{2}\biggl\{ -(d^2 + x^2 + y^2) \pm \biggl[ (d^2 + x^2 + y^2)^2 + 4d^2 x^2 \biggr]^{1 / 2} \biggr\}$

Only the superior — that is, only the positive — sign will ensure positive values of $~\xi_1^2$, so in summary we have,

 Coordinate Transformation $~\xi_1$ $~=$ $~ \frac{1}{\sqrt{2}}\biggl\{ \biggl[ (d^2 + x^2 + y^2)^2 + 4d^2 x^2\biggr]^{1 / 2} -(d^2 + x^2 + y^2) \biggr\}^{1 / 2} \, ;$ $~\xi_2$ $~=$ $~ \frac{x}{\xi_1} \, ;$ $~\xi_3$ $~=$ $~ z \, .$