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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 intwopoints.

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?

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# Parabolic coordinates

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