world = CountryData["Countries", "SchematicPolygon"];
swap[c_] := {c[[2]], -(c[[1]] - 90)};
tmp = Flatten[world[[All, 1, 1]], 1];
tmp = Join[tmp, CountryData[\!\(\*
NamespaceBox["LinguisticAssistant",
DynamicModuleBox[{Typeset`query$$ = "antartica", Typeset`boxes$$ = 
         TemplateBox[{"\"Antarctica\"", 
RowBox[{"Entity", "[", 
RowBox[{"\"GeographicRegion\"", ",", "\"Antarctica\""}], "]"}], 
           "\"Entity[\\\"GeographicRegion\\\", \\\"Antarctica\\\"]\"",
            "\"geographic region\""}, "Entity"], 
         Typeset`allassumptions$$ = {{"type" -> "Clash", 
           "word" -> "antartica", 
           "template" -> "Assuming \"antartica\" is a \
continent${separator}Use as ${desc} or ${desc} instead", 
           "count" -> "3", "pulldown" -> "false", 
           "default" -> "{\"C\", \"antartica\"} -> \
{\"GeographicRegion\", \"dflt\"}", 
           "Values" -> {{"name" -> "GeographicRegion", 
              "desc" -> "a continent", 
              "input" -> "{\"C\", \"antartica\"} -> \
{{\"GeographicRegion\", \"a continent\"}}"}, \
{"name" -> "CountryClass", "desc" -> "a class of countries", 
              
              "input" -> "{\"C\", \"antartica\"} -> \
{\"CountryClass\"}"}, {"name" -> "Desert", "desc" -> "a desert", 
              "input" -> "{\"C\", \"antartica\"} -> {\"Desert\"}"}}}},
          Typeset`assumptions$$ = {}, Typeset`open$$ = {1, 2}, 
         Typeset`querystate$$ = {"Online" -> True, "Allowed" -> True, 
          "mparse.jsp" -> 0.773653`6.340091207439451, 
          "Messages" -> {}}}, 
DynamicBox[ToBoxes[
AlphaIntegration`LinguisticAssistantBoxes["", 4, Automatic, 
Dynamic[Typeset`query$$], 
Dynamic[Typeset`boxes$$], 
Dynamic[Typeset`allassumptions$$], 
Dynamic[Typeset`assumptions$$], 
Dynamic[Typeset`open$$], 
Dynamic[Typeset`querystate$$]], StandardForm],
ImageSizeCache->{225.2265625, {8.11328125, 17.11328125}},
TrackedSymbols:>{Typeset`query$$, Typeset`boxes$$, 
            Typeset`allassumptions$$, Typeset`assumptions$$, 
            Typeset`open$$, Typeset`querystate$$}],
DynamicModuleValues:>{},
UndoTrackedVariables:>{Typeset`open$$}],
BaseStyle->{"Deploy"},
DeleteWithContents->True,
Editable->False,
SelectWithContents->True]\), "SchematicPolygon"][[1, 
    1]]];
world2 = Reap[For[j = 1, j <= Dimensions[tmp][[1]], j++,
      Sow[\[Pi]/180 (swap /@ tmp[[j]])];
      ];][[2, 1]];

ToSphere[c_] := {Cos[c[[1]]] Sin[c[[2]]], Sin[c[[1]]] Sin[c[[2]]], Cos[c[[2]]]};
sphereworld = Reap[For[j = 1, j <= Dimensions[world2][[1]], j++,
      Sow[(ToSphere[#] & /@ world2[[j]])];
      ];][[2, 1]];
ToMercator[c_] := {c[[2]], -Log[Cos[-(c[[1]] - \[Pi]/2)/2] - Sin[-(c[[1]] - \[Pi]/2)/2]] + Log[Cos[-(c[[1]] - \[Pi]/2)/2] + Sin[-(c[[1]] - \[Pi]/2)/2]]};
sinstep[t_] := Sin[\[Pi]/2 t]^2
frames = Table[
   m = RotationTransform[-2 \[Pi] sinstep[t], {0, 1., 0}];
   rotatedsphereworld = 
    Reap[For[j = 1, j <= Dimensions[sphereworld][[1]], j++,
        Sow[(m /@ sphereworld[[j]])];
        ];][[2, 1]];
   mercatorworld = 
    Reap[For[j = 1, j <= Dimensions[rotatedsphereworld][[1]], j++,
        Sow[(ToMercator[#] & /@ (CoordinateTransform["Cartesian" -> "Spherical", rotatedsphereworld[[j]]][[All, 2 ;; 3]]) )];
        ];][[2, 1]];
   mercatorworld2 = 
    Reap[For[j = 1, j <= Dimensions[mercatorworld][[1]], j++,
        Sow[Split[mercatorworld[[j]], Abs[#2[[1]] - #1[[1]]] < 1 &]
          ];
        ];][[2, 1]]; 
   GraphicsRow[{
     Graphics3D[{Thick, Line /@ rotatedsphereworld, LightGray, Sphere[{0, 0, 0}, 1]}, Lighting -> "Neutral", Boxed -> False, ViewPoint -> {2, 0, 0}]
     ,
     Graphics[{Line /@ mercatorworld2, Thick, Line[1.01 {{-\[Pi], -\[Pi]}, {-\[Pi], \[Pi]}, {\[Pi], \[Pi]}, {\[Pi], -\[Pi]}, {-\[Pi], -\[Pi]}}]}, PlotRange -> 1.*{{-\[Pi], \[Pi]}, {-\[Pi], \[Pi]}},       AspectRatio -> 1/1.61803, ImageSize -> 500]
     }]
   , {t, 0, 1, 0.01}];
ListAnimate[frames]