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

Here's the simplest idea I could come up with (thanks to belisarius for some of the code).

• Project the inequalities onto the sphere using spherical coordinates (with θ=q, φ=f).
• Plot these as a flat region plot.
• Then plot this as a texture the sphere.

Here's a couple of homogeneous inequalities of order 3

ineq = {x^3 < x y^2, y^2 z > x z^2};

coords = {x -> r Sin[q] Cos[f], y -> r Sin[q] Sin[f], z -> r Cos[q]}/.r -> 1

region = RegionPlot[ineq /. coords, {q, 0, Pi}, {f, 0, 2 Pi}, 
  Frame -> None, ImagePadding -> 0, PlotRangePadding -> 0, ImageMargins -> 0]

ParametricPlot3D[coords[[All, 2]], {q, 0, Pi}, {f, 0, 2 Pi}, 
 Mesh -> None, TextureCoordinateFunction -> ({#4, 1 - #5} &), 
 PlotStyle -> Texture[Show[region, ImageSize -> 1000]]]