Cutting a section from a sphere by planes

Question

I want to plot and find out the section of a sphere remaining after putting constraints in terms of cartesian planes. How it can be done?

For example, if I have a sphere of $r = 1$, and I put the constraint $z > 0$ on it, simple imagination would suggest that it leaves half the sphere as defined by limits $\theta\in [0, \pi/2]$ and $\phi\in [0, 2\pi]$. Adding the constraint $x > 0$ leaves a quarter of the sphere with $\theta\in [0, \pi/2]$ and $\phi,\in [0, \pi]$, and so on.

I want to achieve the same using Mathematica. I want find the $(\phi, \theta)$ limits for more complex Cartesian constraints applied simultaneously, such as $x > y \land z > y$.

Will appreciate any suggestions.

Answer 1

ClipPlanes does exactly what you need:

You can use ClipPlanes in two different ways:

(1) As an option to clip all the 3D primitives:

Graphics3D[{Green, Sphere[{0, 0, 0}, 1/4], Blue, Sphere[{0, 0, 0}, 2/3], Red, Sphere[]}, 
    ClipPlanes -> {{0, 1, -1, 0}}, 
    ClipPlanesStyle -> Opacity[.25, Gray]]

(2) As a directive that applies to individual 3D primitives:

Graphics3D[{Green, Sphere[{0, 0, 0}, 1/4], 
  ClipPlanes -> {{0, 1, -1, 0}}, Blue, Sphere[{0, 0, 0}, 2/3], 
  ClipPlanes -> {{0, 0, 1, 0}, {0, 1, 0, 0}}, Red, Sphere[]}]

Answer 2

{x, y, z} = {Sin[θ] Cos[φ], Sin[θ] Sin[φ], Cos[θ]};

Reduce[0 <= θ <= π && 0 <= φ < 2π && z > 0, {θ, φ}]

0 <= θ < π/2 && 0 <= φ < 2π

Reduce[0 <= θ <= π && 0 <= φ < 2π && z > 0 && x > 0, {θ, φ}]

0 < θ < π/2 && (0 <= φ < π/2 || 3π/2 < φ < 2π)

Reduce[0 <= θ <= π && 0 <= φ < 2π && z > y, {θ, φ}] // FullSimplify

(θ >= 0 && 4θ < π && 0 <= φ < 2π) || (4θ == π && ((φ >= 0 && 2φ < π) || π/2 < φ < 2π)) || (π/4 < θ < π/2 && (0 <= φ < ArcSin[Cot[θ]] || (φ + ArcSin[Cot[θ]] > π && φ < 2π))) || (π/2 <= θ < 3π/4 && φ + ArcSin[Cot[θ]] > π && φ < 2π + ArcSin[Cot[θ]])

For a graphical solution (to get a quick idea) you can do

RegionPlot[x > y && z > y, {φ, 0, 2π}, {θ, 0, π},
  AspectRatio -> Automatic, FrameLabel -> {φ, θ}]

Answer 3

SphericalPlot3D[{1, 2, 3}, 
{θ, 0, π}, 
{φ, 0, 3 π/2}]

or

Manipulate[
 SphericalPlot3D[1,
  {θ, 0, θf},
  {φ, 0, φf},
  PlotRange -> {{-1, 1}, {-1, 1}, {-1, 1}}],
 {{θf, π/2}, 0, π},
 {{φf, π}, 0, 2 π}
 ]