Wolfram Cloud Document

With[{wm=2ArcTanh[Tan[ArcSinh[1]/2]]},pts2=Table[With[{v=2ArcTan[Tanh[w/2]]},{Cos[u]Cosh[v],Cosh[v]Sin[u],-IEllipticE[Iv,-1]}//N],{u,0,2Pi,Pi/20},{w,-wm,wm,wm/6}]];

With[{wm=2ArcTanh[Tan[ArcSinh[1]/2]]},pts3=Table[With[{v=2ArcTan[Tanh[w/2]]},{Cos[u]Cosh[v],Cosh[v]Sin[u],-IEllipticE[Iv,-1]}//N],{u,Pi/40,2Pi,Pi/20},{w,-11/12wm,wm,wm/6}]];

weave[f_,out_,in_]:={MapThread[f,{Drop[out,-1],in}],MapThread[f,{Drop[out,1],in}]}

Show[Graphics3D[{Line/@pts3,Line/@pts2,weave[weave[Line[{##}]&,#1,#2]&,pts2,pts3]}]];

Pseudosphere; curvature -1

ParametricPlot3D[{Cos[u]Sin[v],Sin[u]Sin[v],Cos[v]+Log[Tan[v/2]]},{u,0,2Pi},{v,0.01,1}];

Kuen surface

ParametricPlot3D[{2(Cos[u]+uSin[u])Sin[v]/(1+u^2Sin[v]^2),2(Cos[u]-uSin[u])Sin[v]/(1+u^2Sin[v]^2),Log[Tan[v/2]]+2Cos[v]/(1+u^2Sin[v]^2)},{u,-4,4},{v,0.01,Pi-0.1},PlotPoints->{60,15}];