Cfxtrjtrk

I have the following triangle in TikZ MWE:

documentclass[tikz]{standalone}

usepackage{pgfplots,mathtools} 

usetikzlibrary{hapes,decorations.pathreplacing}  

usetikzlibrary{patterns}

definecolor{RoyalAzure}{rgb}{0.0, 0.22, 0.66}  



begin{document}



begin{tikzpicture}

draw[pattern color=black!50!white,pattern=dots, line width=0.6pt] (0,0) -- (2,3.4641) -- (4,0)--cycle;

end{tikzpicture}



end{document}

that generates:

I would like to project this triangle to the surface of a sphere, much like this figure:

How can I do this?

The angles of the triangle on the sphere are 3 times 90 degrees whereas the angles of the triangle in the plane are 60 degrees each. Therefore I do not precisely understand what is meant by "project". If it is meant that the triangle on the sphere should also have three equal angles, you could do e.g.

documentclass[tikz,border=3.14mm]{standalone}

usepackage{tikz-3dplot}

usetikzlibrary{patterns,backgrounds}

begin{document}

tdplotsetmaincoords{70}{30}

begin{tikzpicture}[tdplot_main_coords,declare function={R=pi;}]

 shade[tdplot_screen_coords,ball color=gray,opacity=0.5] (0,0) coordinate(O)

  circle[radius=R];

 draw plot[variable=x,domain=tdplotmainphi-180:tdplotmainphi,smooth]

 ({R*cos(x)},{R*sin(x)},0);

 draw[blue,pattern=dots,pattern color=blue] 

   plot[variable=x,domain=90:00,smooth] (0,{-R*sin(x)},{R*cos(x)})

    coordinate   (p1) 

 -- plot[variable=x,domain=0:90,smooth] ({R*sin(x)},0,{R*cos(x)})

   coordinate   (p2) 

 -- plot[variable=x,domain=0:90,smooth] ({R*cos(x)},{-R*sin(x)},0)

   coordinate   (p3);

   begin{scope}[on background layer]

    foreach X in {1,2,3}

    { draw[dashed] (O) -- (pX); }

   end{scope}

end{tikzpicture}

end{document}

An alternative could be to use nonlinear transformations to project anything you want on a sphere. We have used this for the Christmas balls in this video (at a time in which the atmosphere were better...). However, when doing this, we run into the above-mentioned problem that the triangle has different angles on the sphere.

documentclass[tikz,border=3.14mm]{standalone}

usetikzlibrary{patterns}

usepgfmodule{nonlineartransformations}

makeatletter

% from https://tex.stackexchange.com/a/434247/121799

tikzdeclarecoordinatesystem{sphere}{

    tikz@scan@one@pointrelax(#1)

    spheretransformation

}

% 

defspheretransformation{% similar to the pgfmanual section 103.4.2

pgfmathsincos@{pgf@sys@tonumberpgf@x}%

pgfmathsetmacro{relX}{thepgf@x/28.3465}%

pgfmathsetmacro{relY}{thepgf@y/28.3465}%min(max(

pgfmathsetmacro{myx}{28.3465*Radius*cos(min(max((relY/Radius)*(180/pi),-90),90))*sin(min(max((relX/Radius)*cos(min(max((relY/Radius)*(180/pi),-90),90))*(180/pi),-90),90))}

pgfmathsetmacro{myy}{28.3465*Radius*sin(min(max((relY/Radius)*(180/pi),-90),90))}%typeout{(relX,relY)->(myx,myy)}%

pgf@x=myx pt%

pgf@y=myy pt%

} 

makeatother

begin{document}

begin{tikzpicture}[pics/trian/.style={code={

draw[pattern color=black!50!white,pattern=dots, line width=0.6pt] (0,0) -- (2,3.4641) -- (4,0)--cycle;}}]

 pgfmathsetmacro{Radius}{4}

 shade[ball color=red] (0,0) circle[radius=Radius];

 begin{scope}[xshift=-10cm]

  path (0,0) pic{trian};

 end{scope}

 begin{scope}[transform shape nonlinear=true]

  pgftransformnonlinear{spheretransformation}

  pic[local bounding box=box1] at (0,0) {trian};

 end{scope}

end{tikzpicture}

end{document}