Intersection of two cylinders

c1[t_,θ_]:={t,Sin[θ],-Cos[θ]};
c2[s_,θ_]:={Sin[θ],s,-Cos[θ]};

Manipulate[
Show[{Graphics3D[Polygon[{{(Sqrt[1-(h^2)]),(Sqrt[1-(h^2)]),h},{-(Sqrt[1-(h^2)]),(Sqrt[1-(h^2)]),h},{-(Sqrt[1-(h^2)]),-(Sqrt[1-(h^2)]),h},{(Sqrt[1-(h^2)]),-(Sqrt[1-(h^2)]),h}}]],
ParametricPlot3D[{c1[t,θ],c2[t,θ]},{t,-tMax,tMax},{θ,-θMax,θMax}]
},PlotRange->{{-2,2},{-2,2},{-1,1}},Ticks->False,Boxed->False,Axes->False
],{{tMax,2,"length"},0,2},{{θMax,π,"θ"},0.1,π},{h,-1,1}]

Rotation about various lines

f[x_]:=((2(Sqrt[x]))/(1+(x^2)));

Manipulate[ParametricPlot3D[{t Cos[θ],t Sin[θ], u f[t]+2(1-u)},{t,0,1},{u,0,1},PlotRange->{{-1,1},{-1,1},{0,2}}],{θ,0,2π}]

Manipulate[ParametricPlot3D[{t, (u f[t]+2(1-u))Cos[θ], (u f[t]+2(1-u)) Sin[θ]},{t,0,1},{u,0,1},PlotRange->{{0,1},{-2,2},{-2,2}}],{θ,0,2π}]

Manipulate[ParametricPlot3D[{t, (5-(u f[t]+2(1-u)))Cos[θ]+5, (5-(u f[t]+2(1-u))) Sin[θ]+5},{t,0,1},{u,0,1},PlotRange->{{0,1},{0,10},{0,10}},BoxRatios->1],{θ,0,2π}]