Gaussian Curvature

Manipulate[
 a = If[t < 0.5, -1 + 2 t, If[t < 0.6, 0, If[t > 0.6, 1.1 t - 0.6]]];
 Show[Plot3D[{Sqrt[1 + If[a <= 0, a, -a] x^2 + a y^2], 1}, {x, -2,
    2}, {y, -2, 2}, PlotRange -> {0, 2}, Boxed -> False, Axes -> None,
    PlotStyle -> {Directive[Yellow, Specularity[White, 20],
      Opacity[0.8]],
     Directive[Red, Specularity[White, 20], Opacity[0.8]]},
   Mesh -> {2, 4}, ViewPoint -> {5 Sin[t/2], 5 Cos[t/2], 2}],
  Graphics3D[{PointSize[0.02], Thick,
    If[a < 0, {Black,
      Text[Style["Positive Curvature", Large], {0, 0, 2}], Blue,
      Point[{0, 0, 1.05}]},
     If[a == 0, {Black,
       Text[Style["Zero Curvature", Large], {0, 0, 2}], Blue,
       Polygon[{{-2, -2, 1}, {-2, 2, 1}, {2, 2, 1}, {2, -2, 1}}]},
      If[a > 0, {Black,
        Text[Style["Negative Curvature", Large], {0, 0, 2}], Blue,
        Line[{{-2, -2, 1.05}, {2, 2, 1.05}}],
        Line[{{-2, 2, 1.05}, {2, -2, 1.05}}]}]
      ]]
    }]
  ], {t, 0, 1}]