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I want to plot an equation in elliptical coordinates. In the traditional elliptical system, Ellipse and hyperbola shares the same confocal point. Is there a plot function that draws a function in this system? I do not want to parameterized in regular Cartesian. This will defeat the purpose. For example. For example, having nu as the elliptical coordinate and Theta as the Hyperbolic coordinate, how I plot the hyperbolic function

nu=3(Theta)-2

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    $\begingroup$ What does it mean " This will defeat the purpose"? What is the "purpose"? $\endgroup$ – Dr. belisarius Nov 8 '14 at 14:16
  • $\begingroup$ Defeat the purpose of having a different coordinate system still linked to the cartesian coordinates. $\endgroup$ – Giovanni F. Nov 8 '14 at 16:22
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    $\begingroup$ The meaning of the term "elliptical coordinate system" is that it provides a parametrization of points in a Cartesian frame, where every point is the intersection of an ellipse and a hyperbola. So to plot in this coordinate system is synonymous to forming Cartesian plot in the end, and therefore it is not clear at all what you mean by not having elliptical coordinates linked to the Cartesian ones. The name "elliptical coordinates" itself comes from the elliptical shape of the coordinate lines when drawn in Cartesian coordinates! $\endgroup$ – Jens Nov 8 '14 at 20:02
  • $\begingroup$ Explicit equations to transform from Cartesian to elliptic coordinates are given below researchgate.net/publication/… $\endgroup$ – Che Sun Feb 26 '17 at 6:43
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I may have "defeated the purpose":

f[u_, v_, c_] := c {Cos[u], Sin[u]} {Cosh[v], Sinh[v]}
Show[ParametricPlot[{f[u, 3 u - 2, 1], f[u, 3 u - 2, -1]}, {u, 0, 
   2 Pi}, PlotRange -> {{-6, 6}, {-6, 6}}, 
  PlotStyle -> Directive[Red, Thickness[0.01]]],
 ParametricPlot[{f[u, v, 1], f[u, v, -1]}, {u, 0, 2 Pi}, {v, 0, 5}, 
  MeshFunctions -> {#3 &, #4 &}, 
  Mesh -> {Range[0, 2 Pi, 0.3], Range[0, 2 Pi, 0.3]}, 
  MeshStyle -> {Purple, Orange}, PlotRange -> {{-6, 6}, {-6, 6}}, 
  PlotStyle -> {LightBlue, LightBlue}]]

enter image description here

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I'm confused by the "defeat the purpose" statement, but here's a shot. I'm well out of my league here, and tossing this answer out since I was interested in learning something about coordinate transformations in Mathematica.

tr = CoordinateTransformData["Elliptic" -> "Cartesian", 
   "Mapping", {nu, theta}] /. nu -> 3 theta - 2
ParametricPlot[% /. \[FormalA] -> 1, {theta, 0, Pi}]

enter image description here

I'm unsure how to handle the Formal A.

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