plotting a function in bipolar coordinates

How to plot a function specified in bipolar coordinates if straightforward substitution of expression of Cartesian ones via them is impossible?

• "a function specified in bipolar coordinates" - do you have an example on hand? May 13 '16 at 9:11
• Analogous solution should work: graph in the Polar Plane
– Kuba
May 13 '16 at 9:13
• Yes, the function I wish to see is Sqrt[Cosh[u] - Cos[v]]/Sinh[u] \!( (*SubsuperscriptBox[(f), ((-1)/2), (,)])[(Cosh[u])]) where Subscript[f, -1/2] is Legendre' s either P or Q of order - 1/2 of imaginary argument. Too complcated for a beginner. May 14 '16 at 16:20
• you should edit the question with your function. May 14 '16 at 17:40

We can do this by implementing the transformation formula directly: (ref https://en.wikipedia.org/wiki/Bipolar_coordinates )

bipolar[a_] =  a {Sinh[#[]], Sin[#[]]}/
(-Cos[#[]] + Cosh[#[]]) &

Show[{ParametricPlot[
Table[bipolar@{s, t}, {t,
Cases[Range[-3, 3], Except]}] , {s, -Pi, Pi},
AspectRatio -> Automatic, PlotRange -> All],
ParametricPlot[
Table[bipolar@{s, t}, {s,
Cases[Range[-3, 3], Except]}] , {t, -Pi, Pi},
AspectRatio -> Automatic, PlotRange -> All]}] or use the built in CoordinateTransform

ParametricPlot[
Evaluate[ CoordinateTransform[
{{"Bipolar", {1}} -> "Cartesian"}, {s, 1}]] ,
{s, -Pi, Pi}, AspectRatio -> Automatic, PlotRange -> All]]

(note Evaluate is essential here or it will be extremely slow)

• Thank you very much. I try to understand what is written. May 14 '16 at 16:21