Vtyjkyuk

I'm trying to display two highly zoomed-in spheres with the following code:

src = -49.276947, -7.02026463, 8334.27539;
R = 6371000;
Ha = 50000;
Graphics3D[Point[src], Opacity[0.1], Green, Sphere[0, 0, -R, R], 
 Lighter@Blue, Sphere[0, 0, -R, R + Ha], 
 PlotRange -> (src + -R/10, R/10, -R/10, R/10, -2 Ha, 2 Ha)]

As you can see in the output, the sphere mesh is quite coarse. How can I make it finer, so that the image I get really looked like containing spheres, not some connected triangles?

There's Method -> "SpherePoints" -> n:

src = -49.276947, -7.02026463, 8334.27539;
R = 6371000;
Ha = 50000;
Graphics3D[Point[src], Opacity[0.1], Green, Sphere[0, 0, -R, R], 
 Lighter@Blue, Sphere[0, 0, -R, R + Ha], 
 Method -> "SpherePoints" -> 100, 
 PlotRange -> (src + -R/10, R/10, -R/10, R/10, -2 Ha, 2 Ha), 
 ViewPoint -> Front]

You can obtain a piece of the sphere in customizable discretization with

DiscretizeRegion[
 RegionIntersection[
 Cuboid @@ 
 Transpose[src + -R/10, R/10, -R/10, R/10, -2 Ha, 2 Ha],
 Sphere[0, 0, -R, R]
 ],
 MaxCellMeasure -> 1 -> R/600
 ]

You can get the underlying GraphicsComplex with

GraphicsComplex[
 MeshCoordinates[S],
 EdgeForm, MeshCells[S, 2, "Multicells" -> True]
 ]

You can make a smoother sphere with NURBS.

ClearAll@nurbsSphere;
nurbsSphere[c : _?NumberQ, _, _, r_: 1] :=
 Module[
 base = r 
 0,0,-1, 0,0,-1, 0,0,-1, 0,0,-1, 0,0,-1, 0,0,-1, 0,0,-1,
 0,-1,-1, 1,-1,-1, 1,1,-1, 0,1,-1, -1,1,-1, -1,-1,-1, 0,-1,-1,
 0,-1,1, 1,-1,1, 1,1,1, 0,1,1, -1,1,1, -1,-1,1, 0,-1,1,
 0,0,1, 0,0,1, 0,0,1, 0,0,1, 0,0,1, 0,0,1, 0,0,1
 /. p : _?NumberQ, _, _ :> p + c,
 weights = 1, 0.5, 0.5, 1, 0.5, 0.5, 1,
 knots = 0, 0, 0, 1/4, 1/2, 1/2, 3/4, 1, 1, 1
 ,
 BSplineSurface[base, SplineDegree -> 2, 
 SplineKnots -> Automatic, knots, 
 SplineWeights -> weights, 0.5 weights, 0.5 weights, weights, 
 SplineClosed -> False, True]
 ]

Use it just like a normal Sphere:

Graphics3D@nurbsSphere@0, 0, 0

Graphics3D[Point[src], Opacity[0.1], Green, 
 nurbsSphere[0, 0, -R, R], Lighter@Blue, 
 nurbsSphere[0, 0, -R, R + Ha], 
 PlotRange -> (src + -R/10, R/10, -R/10, R/10, -2 Ha, 2 Ha)]

Although you end up right on the balloon knot.