Plot a revolution surface, with two cross sections to show its shape

Question

I'm having trouble plotting a revolution surface with intersections. What I have done is shown in the figure, with the help of SphericalPlot3D: .

But I can't plot the two cross-sections to make it an enclosed surface. And the final style I like to achieve is like the figure below:

i.e., I'd like to have enclosed surface, with different types of meshes if possible, moreover, I wonder if I can plot something without color of the surface, just like the desired style?

Below are the code I used to generate figure 1:

r = -16/(5 π) (θ - 0.5 π)^2 + π;
s1 = SphericalPlot3D[{r},
       {θ, 0, Pi}, {ϕ, 0, 3.2 π/2}, 
       PlotStyle -> Directive[White, Opacity[1]],
       PlotPoints -> 30, 
       Boxed -> False,
       AxesOrigin -> {0, 0, 0}, 
       TicksStyle -> Directive[FontOpacity -> 0, FontSize -> 0], 
       AxesStyle -> Directive[Black, 12], 
       Mesh -> {Range[-π, π, 0.04 π], Range[0, 2 π, 2 π], {}}, 
       ImageSize -> 1000, 
       PlotRange -> {{-1.5 π, 1.5 π}, {-1.5 π, 1.5 π}, {-1 π, 1 π}}
      ]

Thanks a lot!

Answer 1

With addition to your SphericalPlot3D you can trnsform the equation and plot the boudary with ParametricPlot3D.

Show[ (*your plot with fixed lighting*)
 SphericalPlot3D[{-16/(5 \[Pi]) (\[Theta] - 0.5 \[Pi])^2 + \[Pi]}, 
   {\[Theta], 0, Pi}, {\[Phi], 0, 3.2 \[Pi]/2}, 
   PlotStyle -> Directive[White, Opacity[1]], 
   PlotPoints -> 30, Boxed -> False, AxesOrigin -> {0, 0, 0}, 
   TicksStyle -> Directive[FontOpacity -> 0, FontSize -> 0], 
   AxesStyle -> Directive[Black, 12], 
   Mesh -> {Range[-\[Pi], \[Pi], 0.04 \[Pi]], Range[0, 2 \[Pi], 2 \[Pi]], {}}, 
   ImageSize -> 1000, Lighting -> {{"Ambient", White}}
 ],
 ParametricPlot3D[ (*boundaries*)
   Evaluate[ CoordinateTransform[
      "Spherical" -> "Cartesian", 
      {(-16/(5 \[Pi]) (\[Theta] - 0.5 \[Pi])^2 + \[Pi]) rr, \[Theta], #}]],
      {\[Theta], 0, Pi}, {rr, 0, 1}, 
   PlotStyle -> White, 
   MeshFunctions -> {# &, #2 &, #3 &}, 
   Lighting -> {{"Ambient", White}}
 ] & /@ {-.4 \[Pi], 0}
]