GeoGraphics Day/Night way off

Question

I've been struggling to figure out the correct GeoGraphics options to get the desired results. Here's what I have so far and the results that I'm getting:
   g = Show[
   Graphics[{}, ImageSize -> {Automatic, 1000}],
   GeoGraphics[GeoRange -> "World", GeoCenter -> {60, 0}, 
    GeoProjection -> "Orthographic", 
    GeoGridLines -> Quantity[20, "AngularDegrees"], 
    GeoGridLinesStyle -> {AbsoluteThickness[1], 
      AbsoluteThickness[1]},
    GeoBackground -> {"CountryBorders", "Land" -> LightGray, 
      "Ocean" -> White, "Border" -> Black}
    ],
   GeoGraphics[{GeoStyling[Opacity[0.1]], 
     EdgeForm[{Black, AbsoluteThickness[7]}], 
     DayHemisphere[{2020, 01, 01, 15}]}]
   ];
   Magnify[g, 0.5]

And here's the result:

Rather than being the expected hemisphere, it's a wacky shape.

Chip Hurst · Accepted Answer

I think what's causing the issue is that the hemisphere polygon boundary is being folded over itself. One such workaround is to triangulate this polygon.
We can convert DayHemisphere to Polygon by going under the hood with GeoGraphics`GeoEvaluate, and then to a fine union of triangles with DiscretizeRegion.
tris = DiscretizeRegion[
  GeoGraphics`GeoEvaluate[DayHemisphere[{2020, 01, 01, 15}]], 
  MaxCellMeasure -> 10];

MeshCellCount[tris]

{4168, 11275, 7108}

p = MeshPrimitives[tris, 2][[All, 1]];

gp = Apply[{Mod[#1, 180, -90], Mod[#2, 360, -180]} &, Reverse[p, {3}], {2}];

GeoGraphics[{GeoStyling[{Red, Opacity[0.5]}],  
  {EdgeForm[], Polygon[GeoPosition[gp]]}, 
  {AbsoluteThickness[7], DayNightTerminator[{2020, 01, 01, 15}]}}, 
  GeoRange -> "World", GeoProjection -> "Orthographic", GeoCenter -> {60, 0}, 
  GeoGridLines -> Quantity[20, "AngularDegrees"], 
  GeoGridLinesStyle -> {AbsoluteThickness[.5], AbsoluteThickness[.5]},
  GeoBackground -> {"CountryBorders", "Land" -> LightGray, "Ocean" -> White, 
   "Border" -> Directive[Black, AbsoluteThickness[.15]]}]

GeoGraphics Day/Night way off

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