Bsrgvty

I have the following triangle in TikZ MWE:

documentclass[tikz]standalone
usepackagepgfplots,mathtools 
usetikzlibraryhapes,decorations.pathreplacing 
usetikzlibrarypatterns
definecolorRoyalAzurergb0.0, 0.22, 0.66 

begindocument

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

enddocument

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
usepackagetikz-3dplot
usetikzlibrarypatterns,backgrounds
begindocument
tdplotsetmaincoords7030
begintikzpicture[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);
 beginscope[on background layer]
 foreach X in 1,2,3
 draw[dashed] (O) -- (pX); 
 endscope
endtikzpicture
enddocument

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
usetikzlibrarypatterns
usepgfmodulenonlineartransformations
makeatletter
% from https://tex.stackexchange.com/a/434247/121799
tikzdeclarecoordinatesystemsphere
 tikz@scan@one@pointrelax(#1)
 spheretransformation

% 
defspheretransformation% similar to the pgfmanual section 103.4.2
pgfmathsincos@pgf@sys@tonumberpgf@x%
pgfmathsetmacrorelXthepgf@x/28.3465%
pgfmathsetmacrorelYthepgf@y/28.3465%min(max(
pgfmathsetmacromyx28.3465*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))
pgfmathsetmacromyy28.3465*Radius*sin(min(max((relY/Radius)*(180/pi),-90),90))%typeout(relX,relY)->(myx,myy)%
pgf@x=myx pt%
pgf@y=myy pt%
 
makeatother
begindocument
begintikzpicture[pics/trian/.style=code=
draw[pattern color=black!50!white,pattern=dots, line width=0.6pt] (0,0) -- (2,3.4641) -- (4,0)--cycle;]
 pgfmathsetmacroRadius4
 shade[ball color=red] (0,0) circle[radius=Radius];
 beginscope[xshift=-10cm]
 path (0,0) pictrian;
 endscope
 beginscope[transform shape nonlinear=true]
 pgftransformnonlinearspheretransformation
 pic[local bounding box=box1] at (0,0) trian;
 endscope
endtikzpicture
enddocument