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

Please look into RegionFunction. You can use your inequalities verbatim in it inside ParametricPlot3D.

Show[{ParametricPlot3D[{Sin[th] Cos[ph], Sin[th] Sin[ph], 
    Cos[th]}, {th, 0, Pi}, {ph, 0, 2 Pi}, 
   RegionFunction -> 
    Function[{x, y, z}, And[x^3 < x y z + z^3, y^2 z < y^3 + x z^2]], 
   PlotRange -> {-1, 1}, PlotStyle -> Red], 
  Graphics3D[{Opacity[0.2], Sphere[]}]}]