Clear[x,y,z,R0,r,th,ph]
x=(R0+r*Cos[th])*Cos[ph];
y=(R0+r*Cos[th])*Sin[ph];
z=r*Sin[th];
Manipulate[ParametricPlot3D[{x,y,z}/.{R0-> 1,r->1/A},{th,-Pi,Pi},{ph,0,2*Pi}],{{A,3},1.1,5}]

nutilde[th_,a_]:=-(( th+2 ArcTan[(((a-1) Tan[(th/2)])/(Sqrt[1-a^2]))])/(Sqrt[1-a^2]))
Manipulate[Plot[nutilde[th,1/A],{th,-Pi,Pi}],{{A,3},1.1,5}]

thf[th_,a_]:=th+2*a*(th+2 ArcTan[(((a-1) Tan[(th/2)])/(Sqrt[1-a^2]))])/(1-a^2)
Manipulate[Plot[thf[th,1/A],{th,-Pi,Pi}],{{A,5},1.01,10}]

Manipulate[ContourPlot[
thf[ArcTan[x,y],1/A],{x,-1,1},{y,-Sqrt[1-x^2],Sqrt[1-x^2]},ImageSize->{300,300},AspectRatio->Full,Contours->12],{{A,5},2.01,10}]

phiofth[th_,a_]:=-2 ArcTan[(((a-1) Tan[(th/2)])/(Sqrt[1-a^2]))]

Manipulate[Plot[Mod[q*phiofth[th,1/A]-Pi,2*Pi]-Pi,{th,-Pi,Pi},PlotRange->{ -Pi,Pi},AxesLabel->{theta,phi}],{{A,3},1.1,10},{{q,2.5},0.1,5}]

Manipulate[plot1=ParametricPlot3D[{x,y,z}/.{R0-> 1,r->1/A},{th,-Pi,Pi},{ph,0,2*Pi},Mesh->None,PlotStyle-> {Opacity[.3]}];plot2=ParametricPlot3D[{x,y,z}/.{R0-> 1,r->1/A,ph-> q*phiofth[th,1/A]},{th,-Pi,Pi},PlotStyle-> Thick];
Show[plot1,plot2],{{A,3},1.1,10},{{q,2.5},1,10}]

Manipulate[plot1c=ParametricPlot3D[{x,y,z}/.{R0-> 1,r->1/A},{th,-Pi,Pi},{ph,0,2*Pi},Mesh->None,PlotStyle-> {Opacity[.3]}];plot2c=ParametricPlot3D[{x,y,z}/.{R0-> 1,ph-> q*phiofth[th,r]},{th,-Pi,Pi}, {r,0,1/A}];
Show[plot1c,plot2c],{{A,3},1.1,10},{{q,2.5},1,10}]