Conversion from 2-dimensional parabolic coordinates to cartesian and cylindrical I have been looking at the Wolfram Mathworld page on parabolic coordinates here: http://mathworld.wolfram.com/ParabolicCoordinates.html and I'm having trouble grasping how to convert between parabolic and cartesian/polar coordinates when you only want to work in 2 dimensions.  As these equations say: 
$$u=\sqrt{\sqrt{x^2+y^2+z^2}+z}$$ and $$v=\sqrt{\sqrt{x^2+y^2+z^2}-z}$$
So, when you are only working in the xy plane, can you say that u and v are equal to each other?  I am confused because a text I am reading defines u and v, with respect to cylindrical coordinates as: $ u  = \sqrt{r+z} $ and $ v = \sqrt{r-z} $ which clearly aren't equal to each other. 
Thanks for the help!
 A: Since $r=\sqrt{x^2+y^2+z^2},$ MathWorld page and your textbook say the same thing. And yes, $z=0$ if and only if $u=v.$
It may help to consider this diagram from
the page http://mathworld.wolfram.com/ParabolicCoordinates.html:

This is a cross-section of the system of coordinates showing the curves
of constant $u$ and constant $v$ in the $y,z$ plane.
One can obtain the complete parabolic coordinate system in $\mathbb R^3$
by rotating the curves in this figure around the $z$ axis,
with the result that the points with a given constant $u$ value
are all on the surface of a paraboloid,
likewise the points with a given constant $v$ value
are all on the surface of a paraboloid.
Select an arbitrary positive constant $c$. 
In the $y,z$ plane, the parabola where $v=c$ is the mirror image 
(across the $y$ axis) of the parabola where $u=c$.
The two parabolas intersect at two points on the $y$ axis.
In the full coordinate system, rotating these figures around the $z$ axis,
we get a paraboloid where $u=c$ and a mirror-image paraboloid where $v = c$,
and the intersection of those paraboloids is a circle in the $x,y$
plane centered at the origin.
The radius of that circle is $u^2 = v^2 = c^2$.
If you choose two different constant values of $u$ and $v$, say
$u = c_1$ and $v=c_2$ (where $c_1\neq c_2$), the two paraboloids
described by $u = c_1$ and $v=c_2$ still intersect in a circle, but
the circle is not in the $x,y$ plane. Instead, the circle of intersection
is in a plane parallel to the $x,y$ plane at some non-zero value of $z$.
In order to describe a specific point in $\mathbb R^3$ in parabolic coordinates,
in addition to the coordinates $u$ and $v$ you need an angle $\theta$,
which is an angle of the rotation that turned the parabolas into paraboloids.
You can describe a point in the $x,y$ plane in polar coordinates, using
$\theta$ as the angle and $u^2$ 
(or $v^2$, since they are the same in the $x,y$ plane)
as the radius.
If you want curvilinear coordinates in two dimensions that are not
just a reformulation of polar coordinates, you might want to
try parabolic cylindrical coordinates instead 
(http://mathworld.wolfram.com/ParabolicCylindricalCoordinates.html)
and simply delete the $z$ axis.
The resulting coordinate system looks like the figure above,
except that the vertical axis is labeled $x$ instead of $z$.
If you would prefer to have your $x,y$ coordinates in a more
conventional orientation (I would), 
simply reflect the figure through the line $x=y$
(that is, flip it over so the axes are where you want them).
Note that there is an ambiguity in the coordinate system because
the parabolas for $u=u_1$ and $v=v_1$ (for constants $u_1$ and $v_1$)
intersect in two places, one with a positive $y$ coordinate
and one with a negative $y$ coordinate.
As long as you are looking within a region that is all on one side
of the $x$ axis, this is not a problem.
