How to get rid of clipping artifacts?

Question

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?

Answer 1

You can add Method -> {"SpherePoints" -> 300}. With this option there are no artifacts:

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, Method -> {"SpherePoints" -> 300}], Lighting -> "Neutral", 
  ViewPoint -> {1.7, -2.7, 1}, ViewVertical -> {0, 0, 1}, ImageSize -> {1000, Automatic}]

Checked with version 12.0 on Windows 7 x64.