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Parabolic coordinates

  1. Dec 23, 2008 #1
    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: [tex]f(z)=z^{2}[/tex]. What's the difference?
  2. jcsd
  3. Dec 23, 2008 #2


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    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.

    Wikipedia uses two for the two-dimensional case. Mathworld and Wikipedia agree on the three-dimensional case, only they have renamed [tex]u = \tau, v = \sigma, \phi = \theta[/tex]. 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).
    Last edited: Dec 23, 2008
  4. Dec 27, 2008 #3
    I find the complex form here (and also in another book):


    I still have troubles in visualizing how the 1-1 mapping between parabolic and cartesian coordinates works in the 2D case :/
  5. Dec 28, 2008 #4
    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 [tex]u=u_0[/tex] and [tex]v=v_0[/tex], I will have to mark two "points", right?

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

    does this make any sense?
  6. Jun 4, 2011 #5
    I know this post is old but I think my equation shows a link:
    http://dl.dropbox.com/u/13155084/Pythagorean%20lattice.pdf [Broken]


    http://dl.dropbox.com/u/13155084/PL3D2/P_Lattice_3D_2.html [Broken]
    Last edited by a moderator: May 5, 2017
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