t=0.5;Graphics[{
Line[{{-1,-Tan[t]},{5,5 Tan[t]}}],
Circle[{0,0},1,{0,t}],
{AbsolutePointSize[5],Point[z={2.1,2Tan[t]-.1/Tan[t]}],Point[rz={1.9,2Tan[t]+.1/Tan[t]}],Point[zrot={Re[(z[[1]]+I z[[2]])Exp[-I t]],Im[(z[[1]]+I z[[2]])Exp[-I t]]}],Point[rzrot={Re[(rz[[1]]+I rz[[2]])Exp[-I t]],Im[(rz[[1]]+I rz[[2]])Exp[-I t]]}]},
{Dotted,Line[{z,rz}],Line[{zrot,rzrot}]},
Text["α",{0.5,.25Tan[t]}],
Text["z",{2.1,2Tan[t]-.1/Tan[t]},{-3,0}],
Text["r(z)",{1.9,2Tan[t]+.1/Tan[t]},{1.5,0}],
Text["z (E^(-Iα))",{Re[(z[[1]]+I z[[2]])Exp[-I t]],Im[(z[[1]]+I z[[2]])Exp[-I t]]},{-1.3,0}],Text["r(z) (E^(-Iα))",{Re[(rz[[1]]+I rz[[2]])Exp[-I t]],Im[(rz[[1]]+I rz[[2]])Exp[-I t]]},{-1.3,0}]
}
,Axes->True,Ticks->None,AxesLabel->{"","I"},ImageSize->700,BaseStyle->24]

τ=5π/8;Graphics[{Circle[{0,0},1],{AbsolutePointSize[6],Point[a={Cos[τ],Sin[τ]}],Point[b={Sin[τ],-Cos[τ]}],Point[c={-Sin[τ],Cos[τ]}]},{Text["(a,b)",a,{1.3,-.6}],Text["(c,d)=(-b,a)",c,{-1.3,+0.6}],Text["(c,d)=(b,-a)",b,{1.3,-0.6}]},{AbsoluteThickness[2],Line[{{0,0},a}],{Dotted,Line[{{0,0},b}],Line[{{0,0},c}]}}},Axes->True,PlotRange->{{-1.2,1.2},{-1.2,1.2}},BaseStyle->16,Ticks->{None,{1}}]

τ=π/4;{{Cos[τ],Sin[τ]},{-Sin[τ],Cos[τ]}}.{0,1}

τ=π/2;{{Cos[τ],Sin[τ]},{Sin[τ],-Cos[τ]}}.{-1,1}