Is there a buit-in way to get a surface patch of a sphere for Graphics3D?

Question

Mathematica 12.0 for Windows.

Is there a buit-in way to get a surface patch of a sphere for Graphics3D?

In this Graphics3D output, the sphere is only there for backing, and I only need (and want) part of the upper hemisphere under the mesh. The 2D Circle[] function allows me to draw a segment of a circle between angles. I don't see any similar features for Sphere[] in 3D.

I have code that can draw smooth surface patches using parameterized functions, but the requires considerable effort, and often doesn't produce the quality that the builtin Sphere[] function produces. Here's an example.

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Edit to add:

I went ahead and used my surface patch code to produced an approximation of what I'm looking for. It's the backing sphere that I want to trim. For example, this graphic uses a spherical cap.

There are defects in it, such as the divots around the rim, and there is a naval at the pole where my patches share a common vertex.

I'm confident that, with enough effort, I can use my patch function to draw any type of surface patch on a sphere, and remove the flaws. I was hoping for something comparable to the functionality of the 2D Circle[].

Answer 1

You can also use the options MeshFunctions and RegionFunction in SphericalPlot3D to get a patch filled with mesh lines:

{θrange, ϕrange} = π {{1/6, 5/9}, {3 /2, 11/6}};

patch = SphericalPlot3D[1, θ, ϕ, 
  MeshFunctions -> {#4 &, #5 &}, 
  RegionFunction -> (And[Between[θrange]@#4, Between[ϕrange]@#5] &), 
  PlotPoints -> 100, PlotStyle -> Opacity[0], PlotRange -> 1]

Show[patch, Graphics3D[{Opacity[.5], Sphere[]}]]

Answer 2

Clear["Global`*"]

Manipulate[
 Show[
  SphericalPlot3D[1,
   {θ, θint[[1]] Degree, θint[[2]] Degree},
   {ϕ, ϕint[[1]] Degree, ϕint[[2]] Degree},
   PlotRange -> 1.05],
  Graphics3D[{Opacity[0.5], Sphere[{0, 0, 0}, 0.99],
    Darker[Red], Thick, Line[{{0, 0, 0}, #}] & /@
     Flatten[
      Table[{Cos[ϕ] Sin[θ], Sin[θ] Sin[ϕ], 
        Cos[θ]},
       {θ, θint Degree}, {ϕ, ϕint Degree}], 
      1]}],
  AxesLabel -> (Style[#, 14] & /@ {x, y, z})],
 {{θint, {50, 85}, 
   "θ interval (0—180 degrees)"},
  0, 180, 5,
  Method -> "Push",
  MinIntervalSize -> 5,
  ControlType -> IntervalSlider,
  Appearance -> "Labeled"},
 {{ϕint, {270, 330}, "ϕ interval (0—360 degrees)"},
  0, 360, 5,
  Method -> "Push",
  MinIntervalSize -> 5,
  ControlType -> IntervalSlider,
  Appearance -> "Labeled"}]