Mathematica RegionPlot on the surface of the unit sphere?

Question

I am using RegionPlot3D in Mathematica to visualise some inequalities. As the inequalities are homogeneous in the coordinates they are uniquely determined by their

Answer 1

Simon beat me to the punch but here's a similar idea, based on lower level graphics. I deal with linear, homogeneous inequalities of the form Ax>0.

A = RandomReal[{0, 1}, {8, 3}];
eqs = And @@ Thread[
    A.{Sin[phi] Cos[th], Sin[phi] Sin[th], Cos[phi]} >
        Table[0, {Length[A]}]];
twoDPic = RegionPlot[eqs,
    {phi, 0, Pi}, {th, 0, 2 Pi}];
pts2D = twoDPic[[1, 1]];
spherePt[{phi_, th_}] := {Sin[phi] Cos[th], Sin[phi] Sin[th], 
    Cos[phi]};
rpSphere = Graphics3D[GraphicsComplex[spherePt /@ pts2D,
   twoDPic[[1, 2]]]]

Let's compare it against a RegionPlot3D.

rp3D = RegionPlot3D[And @@ Thread[A.{x, y, z} >
      Table[0, {Length[A]}]],
 {x, -2, 2}, {y, -2, 2}, {z, -2, 2},
   PlotStyle -> Opacity[0.2]];
Show[{rp3D, rpSphere}, PlotRange -> 1.4]