Parabolic Coordinates & Cartesian Coordinates - 1-1 Mapping

[mnb96]

Hi,
I have a doubt about parabolic coordinates in 2D.
if u,v are the parabolic coordinates in a plane, and we keep v=v0 constant, we have a parabola. Analogously keeping u=u0 we have another parabola which intersect the previous one in two points.

My question is, how there can be a 1-1 mapping between parabolic and cartesian coordinates without introducing a third coordinate?


What confused me is that Mathworld defines parabolic coordinates using 3 coordinates, while in wikipedia you can find a definition which uses only two coordinates and an elegant form using complex numbers: $$f(z)=z^{2}$$. What's the difference?

 

[CompuChip]

My question is, how there can be a 1-1 mapping between parabolic and cartesian coordinates without introducing a third coordinate?

As far as I understand, there is a 1-1 mapping between two-dimensional parabolic coordinates (in the Wikipedia notation: tau, sigma) and two-dimensional cartesian coordinates (x, y). Also, there is one between three-dimensional parabolic (tau, sigma, phi) and Cartesian (x, y, z) coordinates.

What confused me is that Mathworld defines parabolic coordinates using 3 coordinates, while in wikipedia you can find a definition which uses only two coordinates and an elegant form using complex numbers: $$f(z)=z^{2}$$. What's the difference?

Wikipedia uses two for the two-dimensional case. Mathworld and Wikipedia agree on the three-dimensional case, only they have renamed $$u = \tau, v = \sigma, \phi = \theta$$. I couldn't find the complex form right away, but remember that complex numbers "are" two-dimensional (there is a 1-1 mapping between complex numbers a + bi and cartesian coordinates (a, b) on the plane).

 

[mnb96]

I find the complex form here (and also in another book):

http://eom.springer.de/P/p071170.htm

I still have troubles in visualizing how the 1-1 mapping between parabolic and cartesian coordinates works in the 2D case :/

 

[mnb96]

ok...let's put my question in this way:

if I am in parabolic coordinates and I want to sketch on paper the intersection between the curves $$u=u_0$$ and $$v=v_0$$, I will have to mark two "points", right?

Instead, if we are in cartesian coordinates the intersection between $$x=x_0$$ and $$y=y_0$$ always yields one point.

does this make any sense?

 

[JeremyEbert]

I know this post is old but I think my equation shows a link:
http://dl.dropbox.com/u/13155084/Pythagorean%20lattice.pdf

and

http://dl.dropbox.com/u/13155084/PL3D2/P_Lattice_3D_2.html