The following is a MWE extracted from a more complex graphic.

`vertex = {8 Sqrt[2/((5 - Sqrt[5]) (10 + 2 Sqrt[5]))], 0, 8/ Sqrt[(10 - 2 Sqrt[5]) (10 + 2 Sqrt[5])]}; example[r_] := Show[Graphics3D[{Opacity[0.3, GrayLevel[0.8]], Sphere[{0, 0, 0}, r], GrayLevel[0.2], Opacity[1], Style[Sphere[{0, 0, 0}, 3], ClipPlanes -> Hyperplane[vertex, Cos[30 Degree]*3*Normalize[vertex]]]}, Boxed -> False], Lighting -> "Neutral", ViewPoint -> {1.7, -2.7, 1}, ViewVertical -> {0, 0, 1}, ImageSize -> {1000, Automatic}] Export["ex1.png", example[3]] Export["ex2.png", example[3.001]] `

The second example is a bit better than the first one, but both show artifacts that I’d like to get rid of. (To clarify what I’m talking about: In both cases, the boundary of the spherical cap is clearly some $ n$ -gon and not a circle. And `Example[2.999]`

would have the same effect.)

How can I improve this? Is there a way to increase the mesh granularity of the sphere? Or is there a better way to create the spherical cap?