mjhjmtu

This is the diagram I would like to recreate. (from https://en.wikipedia.org/wiki/Hyperbolic_triangle)

From Is there any easy way to draw a ruled surface like a hyperbolic paraboloid in TikZ?, I found a way to draw the saddle-shaped surface.

Is there a way to draw the triangle on it? And get the colours/transparency like the picture?

documentclassarticle
usepackagepgfplots
begindocument
begintikzpicture
beginaxis
addplot3 [surf,shader=flat,draw=black] x^2-y^2;
endaxis
endtikzpicture
enddocument 

In principle it is very simple: draw a parametric curve on the manifold and fill it.

documentclassarticle
usepackagepgfplots
pgfplotssetcompat=1.16
begindocument
tikzsetdeclare function=%
fx(x)=ifthenelse(x<0,0.75*(x+1),0.75*(-x+1));
fy(y)=ifthenelse(y<0,0,ifthenelse(y>1,-2+y,-y));

begintikzpicture
beginaxis[view=-2045,axis lines=none,colormap/cool]
addplot3 [surf,shader=interp,draw=black,domain=-1.2:1.2,domain y=-1.5:1.5,opacity=0.6] x^2-y^2;
addplot3 [domain=-2:2,samples=81,smooth,fill=green,fill opacity=0.1] (fx(x),fy(x),fx(x)^2-fy(x)^2);
endaxis
endtikzpicture
enddocument 
begintikzpicture
beginaxis[samples=41]
addplot[domain=-2:2] fx(x);
addplot[blue,domain=-2:2] fy(x);
endaxis
endtikzpicture

UPDATE: Tried to accommodate the requests in your comment. Please note also that the boundaries of the triangle are not pixelated on the pdf, the pixelation comes from the conversion to png.

ADDENDUM: Transparent plot with tikz-3dplot. Note, however, that the top contour is guessed. You can not easily adjust the view angles here without doing some math before.

documentclass[tikz,border=3.14mm]standalone
usepackagetikz-3dplot
usetikzlibraryshadings
usepackagepgfplots
pgfplotssetcompat=1.16
usepgfplotslibraryfillbetween
tikzsetdeclare function=%
f(x,y)=x*x-y*y;
fx(x)=ifthenelse(x<0,0.75*(x+1),0.75*(-x+1));
fy(y)=ifthenelse(y<0,0,ifthenelse(y>1,-2+y,-y));

usetikzlibrarybackgrounds,calc,positioning
begindocument
pgfmathsetmacroxmax1
pgfmathsetmacroymax1.5
foreach X in 190
tdplotsetmaincoords130X 
begintikzpicture[font=sffamily,xscale=4,yscale=2]
%node at (0,0) X;
beginscope[tdplot_main_coords,samples=61,smooth,variable=x]
 draw[name path=boundary] plot[domain=-ymax:ymax] (-xmax,x,f(-xmax,x))
 -- plot[domain=-xmax:xmax] (x,ymax,f(x,ymax)
 -- plot[domain=ymax:-ymax] (xmax,x,f(xmax,x))
 -- plot[domain=xmax:-xmax] (x,-ymax,f(x,-ymax));
 tikzsetdeclare function=ytop(x)=0.35-0.2*(x/xmax);
 draw[name path=top] plot[domain=-xmax:xmax] (x,ytop(x),f(x,ytop(x)));
 shade [%draw,blue,ultra thick,
 top color=blue!80,bottom color=blue,opacity=0.3,
 name path=back,
 intersection segments=
 of=top and boundary,
 sequence=B2--A2[reverse]
 ];
 shade [%draw,blue,ultra thick,
 top color=blue!80,bottom color=blue,opacity=0.3,
 name path=front,
 intersection segments=
 of=top and boundary,
 sequence=B3--B0--A1--A2 
];
 shadedraw[thick,top color=green!20,bottom color=green!40,opacity=0.6] 
 plot[variable=x,domain=-2:2,samples=81] (fx(x),fy(x),fx(x)^2-fy(x)^2);
endscope
endtikzpicture
enddocument