A plane cutting through a cylinder

The plane

Plane=ContourPlot3D[x+z==1,{x,-2,2},{y,-2,2},{z,-2,4}]

The cylinder

Surface=ContourPlot3D[x^2/4+y^2==1,{x,-2,2},{y,-2,2},{z,-2,4}]

The intersection curve

Show[Plane,Surface]

Curve=ParametricPlot3D[{2Cos[t],Sin[t],1-2Cos[t]},{t,0,2Pi},PlotStyle->{Thick,Blue},PlotRange->{{-2,2},{-2,2},{-2,4}}]

Show[Plane,Curve]

Show[Surface,Curve]

Show[Surface,Plane,Curve]

A parametrization of the intersection curve

Animate[ParametricPlot3D[{2Cos[t],Sin[t],1-2Cos[t]},{t,0,n},PlotStyle->{Thick,Blue},PlotRange->{{-2,2},{-2,2},{-2,4}}],{n,0,2Pi}]