ParametricPlot3D[{2 s ((Cos[t])^2),2 s Cos[t] Sin[t],0},{s,0,1},{t,-Pi/2,Pi/2},
PlotRange->{{-2,2},{-2,2},{0,2}}
]

FullSimplify[{x,y,(Sqrt[4-(x^2)-(y^2)])}/.{x->2 s ((Cos[t])^2),y-> 2 s Cos[t] Sin[t]}]

ParametricPlot3D[{2 s ((Cos[t])^2),s Sin[2 t],(Sqrt[2]) (Sqrt[2-(s^2)-(s^2) Cos[2 t]])},{s,0,1},{t,-Pi/2,Pi/2},
PlotRange->{{-2,2},{-2,2},{0,2}}
]

ParametricPlot3D[{2  ((Cos[t])^2),2 s Cos[t] Sin[t],0},{s,-1,1},{t,-Pi/2,Pi/2},
PlotRange->{{-2,2},{-2,2},{0,2}}
]

FullSimplify[{x,y,(Sqrt[4-(x^2)-(y^2)])}/.{x->2  ((Cos[t])^2),y-> 2 s Cos[t] Sin[t]}]

ParametricPlot3D[{2 ((Cos[t])^2),s Sin[2 t],(Sqrt[4-4 ((Cos[t])^4)-4 (s^2) ((Cos[t])^2) ((Sin[t])^2)])},{s,-1,1},{t,-Pi/2,Pi/2},
PlotRange->{{0,2},{-1,1},{0,2}},Mesh->True
]