Loxodrome Shadow

frame[\[Theta]_] := Rasterize@Module[{m, pp, umax, pr},
    m = 0.45;
    pp = 30;
    umax = 0.8;
    range = 2.8;
    pr = {{-range, range}, {-range, range}, {-1.1, 1.1}};
    texture = ParametricPlot[
      proj[
       RotationMatrix[\[Theta] + 0.5, {1, 0, 0}].RotationMatrix[
         u, {0, 0, 1}].loxodrome[\[Lambda], m, 0]]
      , {u, 0, umax}, {\[Lambda], -10, 10}, PlotRange -> 3,
      Mesh -> False,
      PlotStyle -> Directive[RGBColor[0.15, 0.18, 0.5], Opacity[0.5]],
       BoundaryStyle -> None, Frame -> None, Axes -> None,
      PlotPoints -> pp, Background -> RGBColor[0.8, 0.8, 0.9]];
    Show[
     Graphics3D[{RGBColor[0.1, 0.6, 0.7], Opacity[0.2], Sphere[],
       {Opacity[1.0], Texture[texture],
        Polygon[{{-3, -3, -1}, {3, -3, -1}, {3, 3, -1}, {-3, 3, -1}},
         VertexTextureCoordinates -> {{0, 0}, {1, 0}, {1, 1}, {0, 1}}]}
       }, Lighting -> "Neutral", PlotRange -> pr, Boxed -> False],
     ParametricPlot3D[
      {
       RotationMatrix[\[Theta] + 0.5, {1, 0, 0}].RotationMatrix[
         u, {0, 0, 1}].loxodrome[\[Lambda], m, 0]
       }
      , {\[Lambda], -10, 10}, {u, 0, umax}, PlotRange -> pr,
      PlotPoints -> pp, Mesh -> False, PlotStyle -> Darker@Red,
      BoundaryStyle -> None], ImageSize -> 300]
    ];
frame[1.07]