$Version

sue =  \[FormalA]  -> e

suab = {a ->3, b-> 2}

suve = e -> Sqrt[a^2 - b^2] /. suab

sun = Flatten[{sue,suab, suve}]

refell = Circle[{0,0},{a,b}]/.sun

gell =Show[Graphics[ refell]];

Clear[x,y,η,ψ]

{xp,yp} = CoordinateTransformData["Elliptic"->"Cartesian","Mapping",{η,ψ}]/.sue

η0 = Log[(a + b)/(a - b)//N]/2  /.sun

Clear[x,y,η,ψ,ηr,ψr]

fetps[x_,y_] = CoordinateTransformData["Cartesian"->"Elliptic","Mapping",{x,y}]/.sue

tetps = Table[{η,ψ},{η,0,1.6 η0, η0/2.},{ψ,- 1. π,1.π,π/4.}]

Dimensions[tetps]

ListPlot[tetps,PlotStyle -> PointSize[0.01], AxesLabel -> {"η","ψ"}]

txpyp = Table[{xp,yp}/.sun,{η,0,1.6 η0, η0/4.},{ψ,0.,2.π,π/8.}];

Length[txpyp]

pp =Table[ListPlot[txpyp[[k]],PlotStyle ->{ PointSize[0.005],Black}, AxesLabel -> {"x","y"},
AspectRatio -> Automatic,ImageSize ->700],{k,Length[txpyp]}];

Show[gell,pp]

txpyp[[k]]

Table[ListLinePlot[txpyp[[k]],PlotStyle ->{ PointSize[0.005],Black}, AxesLabel -> {"x","y"},
AspectRatio -> Automatic,ImageSize ->700],{k,Length[txpyp]}];

Show[%, PlotRange -> All]

Table[ListLinePlot[txpyp[[k]],PlotStyle ->{ PointSize[0.005],Black}, AxesLabel -> {"x","y"},
AspectRatio -> Automatic,ImageSize ->700],{k,Length[txpyp]}];
Show[%, PlotRange -> All]

Map[(fetps[#[[1]],#[[2]]]/.sun)& ,txpyp]

Dimensions[%]

tetps

Dimensions[%]

txpyp

Dimensions[%]

Table[{η,ψ},{η,0,1.6 η0, η0/2.},{ψ,0.,2.π,π/4.}]

Table[(fetps[xs,ys]/.sun) /.{ xs ->txpyp[[i,k,1]], ys ->txpyp[[i,k,2]]},{i,7},{k,17}]//Chop

tetps