How to get rid of clipping artifacts?

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]] 

First example

Second example

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?

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How to prevent graphics clipping through each other

Consider the following:

g1 = Graphics3D[{Black, Sphere[{0, 0, 0}, 0.1]}]; g2 = Graphics3D[{Directive[Red, Thickness[0.02]], InfiniteLine[{0, 0, 0}, {1, 0, 0}]}]; Show[{g1, g2}, PlotRange -> {{-1, 1}, {-1, 1}, {-1, 1}}, Boxed -> False, ImageSize -> Large] 

When I rotate the resulting 3D image, the red line will clip through the black sphere, despite being smaller than the sphere radius. The edge where they meet appears two dimensional and at some angles the line is completely visible as though it were outside the sphere, see below. How can I prevent this?

enter image description here

enter image description here

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