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I want to draw a triangulation of a sphere starting from the following data. I have a collection of triangles represented by three points $(p1,p2,p3)$ on a sphere of radius 1. I would like to draw in a different color (or maybe even with a texture) each curvy triangle. Any ideas?

EDIT

Thanks for the first answers. Both method works fine, but there is something else that now bothers me.

For each triangle I have a picture that I would like to use as a Texture, each picture has a central "body" surrounded by white empty space that can be increased at pleasure. I would like the Texture to be added so that the central body ends up on the barycenter of the triangle and I would like it not to be too deformed. In other words, what I would like to happen is to cut a triangle out of the original picture and use it as a texture so that the vertices and barycenter (the central body) are mapped to vertices and barycenter of the spherical triangles. Here is an example of a picture

enter image description here

Also, I would like to be able to "see through" the regions, so that one can see the central body drawn on the far side of the sphere without rotating it. If I use PlotStyle->{Texture[Image],Opacity[.5]} from your first Method, the texture completely disappears.

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  • $\begingroup$ The surface of a sphere is two-dimensional. Only two coordinates are needed. You already specified the radius. Or, have you computed already the three dimensional. $\endgroup$ Commented Mar 22, 2023 at 1:23

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Method-1

  • ConicHullRegion[{{0, 0, 0}}, {p1, p2, p3}] generate the infinite pyramid. $$\{ t_1p1+t_2p_2+t_3p_3, t_1,t_2,t_3\geq 0 \}$$

  • The spherical triangle is the intersection of theConicHullRegion region and Sphere[].

SeedRandom[1]; sphere = Sphere[]; {p1, p2, p3} = RandomPoint[sphere, 3]; viewpoint = Mean[{p1, p2, p3}]; reg = DiscretizeRegion[ RegionIntersection[{sphere, ConicHullRegion[{{0, 0, 0}}, {p1, p2, p3}]}], {{-1.2, 1.2}, {-1.2, 1.2}, {-1.2, 1.2}}, MaxCellMeasure -> 10^-5, AccuracyGoal -> 3]; sphericaltriangle = RegionPlot3D[reg, PlotStyle -> Texture[ExampleData[{"ColorTexture", "WhiteMarble"}]], Mesh -> None] Graphics3D[{sphere, sphericaltriangle[[1]]}, ViewPoint -> viewpoint, ViewProjection -> "Orthographic", PlotRange -> All, Boxed -> False]

enter image description here

Method-2

  • For three points{p1,p2,p3}, we parametric the space triangle by
{1 - t, t} . {p1, {1 - s, s} . {p2, p3}}

where 0<=t<=1 and 0<=s<=1. After that we Normalize all of the points of triangle to projecte it to the unit-sphere.

SeedRandom[1]; sphere = Sphere[]; {p1, p2, p3} = RandomPoint[sphere, 3]; ParametricPlot3D[{1 - t, t} . {p1, {1 - s, s} . {p2, p3}} // Normalize, {t, 0, 1}, {s, 0, 1}, PlotStyle -> Texture@ExampleData[{"ColorTexture", "GoldenOak"}], Mesh -> None]

enter image description here

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  • $\begingroup$ The second method is very elegant. And nice use of the texture data! $\endgroup$ Commented Mar 22, 2023 at 13:34

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