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4

Fiddling around...not an answer, other than that some obvious attempts don't work. Try two half annuluses (annuli?) S7 = ImplicitRegion[(1/10 <= x^2 + y^2 <= 1 && (x >= 0 )), {x, y}]; S8 = ImplicitRegion[(1/10 <= x^2 + y^2 <= 1 && (x <= 0 )), {x, y}]; GraphicsGrid[{{RegionPlot[S8 ], RegionPlot[S7]}}] Use ...

3

One possible workaround is to add a thin gap on the annulus: r = TransformedRegion[ Annulus[{0, 0}, {Sqrt[1/10], 1}, {-Pi + 0.001, Pi}], {(#1^2 + #2^2)^0.5, ArcTan[#, #2]} &]; RegionPlot[r, AspectRatio -> Automatic]

7

SeedRandom; dm = RegionBoundary @ DelaunayMesh[RandomPoint[Sphere[], 100]]; mc = MeshCoordinates[dm]; We can use a combination of GeoPositionXYZ and GeoPosition to get 2D projections of mc: gp = Most /@ First @ GeoPosition @ GeoPositionXYZ[mc, 1.]; indices = List /@ MeshCells[dm, 1][[All, 1]]; Row[GeoGraphics[GeoPath[Extract[gp, indices]], GeoRange -&...

12

I wouldn't try to do this on a square (assuming you want at least some sort of systematic correspondence between edges on the sphere and their projections), at least if you don't want crossings on square sides. As @Szabolcs stated, you can't really do this without extreme distortion. You may use map projections, though, picking your poison (for the mesh ...

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