(* Square of the distance between points with angular co-ordinates θ and Ω on a circle of radius SoverR. *)
d2[θ_,Ω_,SoverR_]:=SoverR^2+1-2*SoverR*Cos[Ω-θ]

(* These functions are proportional to (OverHat[ϕ])_1 and (OverHat[ϕ])_2. *)
f1[α_,β_,θ_,ω_,Ω_,SoverR_]:=(α*d2[θ,Ω,SoverR]-β*d2[-θ,Ω,SoverR]-d2[ω,Ω,SoverR])/(α*d2[θ,Ω,SoverR]+β*d2[-θ,Ω,SoverR]+d2[ω,Ω,SoverR])
f2[α_,β_,θ_,ω_,Ω_,SoverR_]:=(α*d2[θ,Ω,SoverR]+β*d2[-θ,Ω,SoverR]-d2[ω,Ω,SoverR])/(α*d2[θ,Ω,SoverR]+β*d2[-θ,Ω,SoverR]+d2[ω,Ω,SoverR])

(* Interactive plot showing the image of the plane under the soliton map, in toric co-ordinates, given partons at angular positions θ, -θ and ω.  The parameters a and b determine the vacuum values of the (OverHat[ϕ])_i.  *)
Manipulate[{Show[ParametricPlot[{f1[a,b,θ,ω,u,s],f2[a,b,θ,ω,u,s]},{u,0,2*Pi},{s,0,1},PlotRange->1,ImageSize->400],Plot[x,{x,-1,1}]]},{{a,1},0.1,10},{{b,1},0.1,10},{{θ,1},0.01,Pi-0.01},{{ω,1},0,2*Pi-0.01}]

GraphicsRow[{Show[ParametricPlot[{f1[1,1,1.6,3.2,u,s],f2[1,1,1.6,3.2,u,s]},{u,0,2*Pi},{s,0,1},PlotRange->1,BaseStyle->{FontFamily->"Helvetica",FontSize->14},Frame->{{True,False},{False,True}},FrameLabel->{{"2(OverHat[(Subscript[ϕ, 2])])/3m",False},{False,"2(OverHat[(Subscript[ϕ, 1])])/3m"}},FrameTicks->{{{-1,0,1},None},{None,{-1,0,1}}},Axes->False,PlotStyle->LightBlue],Plot[x,{x,-1,1}]],Show[ParametricPlot[{f1[1,1,2.8,1,u,s],f2[1,1,2.8,1,u,s]},{u,0,2*Pi},{s,0,1},PlotRange->1,BaseStyle->{FontFamily->"Helvetica",FontSize->14},Frame->{{True,False},{False,True}},FrameLabel->{{"2(OverHat[(Subscript[ϕ, 2])])/3m",False},{False,"2(OverHat[(Subscript[ϕ, 1])])/3m"}},FrameTicks->{{{-1,0,1},None},{None,{-1,0,1}}},Axes->False,PlotStyle->LightBlue],Plot[x,{x,-1,1}]],Show[ParametricPlot[{f1[1,1,Pi,1,u,s],f2[1,1,Pi,1,u,s]},{u,0,2*Pi},{s,0,1},PlotRange->1,BaseStyle->{FontFamily->"Helvetica",FontSize->14},Frame->{{True,False},{False,True}},FrameLabel->{{"2(OverHat[(Subscript[ϕ, 2])])/3m",False},{False,"2(OverHat[(Subscript[ϕ, 1])])/3m"}},FrameTicks->{{{-1,0,1},None},{None,{-1,0,1}}},Axes->False,PlotStyle->LightBlue],Plot[x,{x,-1,1}]]},Spacings->Scaled[0],ImageSize->1200]