I am trying to identify what characteristic distinguishes elliptical coordinates from polar coordinates. For concreteness, let's write down the expressions.
Polar:
$$ x=r \cos(t) \\ y=r \sin(t) $$
Elliptical:
$$ x=a \cosh(r) \cos(t) \\ y=a \sinh(r) \sin(t) $$
(from http://en.wikipedia.org/wiki/Elliptic_coordinate_system)
The most obvious distinction is that the "normal" lines to the ellipses are not straight lines but curved (in fact confocal parabolas).
Another way to say it is that for polar coordinates the normal vector $(dx/dr,dy/dr)$ depends on $t$ only, not on $r$, and is thus normal to all circles (any $r$), whereas for elliptical coordinates the normal vector depends on both $r$ and $t$, so is normal to the ellipse for a specific $r$ and $t$.
So my questions are:
- Is this a good characterization of the difference, and is it possible to state it in a more insightful manner in a larger context?
- Is there a different or better way to characterize the difference?
- Is there a name for it?