Drawing shaded regions bounded by arcs with tikz

Question

I am trying to create a tikz image based on the following screenshot:

I can figure out how to draw a pyramid, but I have so far been hopeless at drawing the red shaded regions. Here is the minimal progress I have made:

\documentclass{article}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}
    [x={(0.648577cm, -0.290453cm)},
    y={(0.759883cm, 0.300370cm)},
    z={(-0.043879cm, 0.908523cm)},
    scale=2.000000,
    back/.style={densely dotted, thin},
    edge/.style={densely dotted, thin},
    facet/.style={fill=blue!95!black,fill opacity=0.000000},
    vertex/.style={inner sep=1pt,circle,draw=green!25!black,fill=green!75!black,thick,anchor=base}]

\coordinate (-0.75000, -0.75000, -0.75000) at (-0.75000, -0.75000, -0.75000);
\coordinate (0.75000, -0.75000, -0.75000) at (0.75000, -0.75000, -0.75000);
\coordinate (-0.75000, 0.75000, -0.75000) at (-0.75000, 0.75000, -0.75000);
\coordinate (0.75000, 0.75000, -0.75000) at (0.75000, 0.75000, -0.75000);
\coordinate (0.00000, 0.00000, 1.00000) at (0.00000, 0.00000, 1.00000);

\fill[facet] (0.00000, 0.00000, 1.00000) -- (-0.75000, -0.75000, -0.75000) -- (-0.75000, 0.75000, -0.75000) -- cycle {};
\fill[facet] (0.00000, 0.00000, 1.00000) -- (-0.75000, -0.75000, -0.75000) -- (0.75000, -0.75000, -0.75000) -- cycle {};
\fill[facet] (0.00000, 0.00000, 1.00000) -- (0.75000, -0.75000, -0.75000) -- (0.75000, 0.75000, -0.75000) -- cycle {};
\fill[facet] (0.00000, 0.00000, 1.00000) -- (-0.75000, 0.75000, -0.75000) -- (0.75000, 0.75000, -0.75000) -- cycle {};

\draw[edge] (-0.75000, -0.75000, -0.75000) -- (0.75000, -0.75000, -0.75000);
\draw[edge] (-0.75000, -0.75000, -0.75000) -- (-0.75000, 0.75000, -0.75000);
\draw[edge] (-0.75000, -0.75000, -0.75000) -- (0.00000, 0.00000, 1.00000);
\draw[edge] (0.75000, -0.75000, -0.75000) -- (0.75000, 0.75000, -0.75000);
\draw[edge] (0.75000, -0.75000, -0.75000) -- (0.00000, 0.00000, 1.00000);
\draw[edge] (-0.75000, 0.75000, -0.75000) -- (0.75000, 0.75000, -0.75000);
\draw[edge] (-0.75000, 0.75000, -0.75000) -- (0.00000, 0.00000, 1.00000);
\draw[edge] (0.75000, 0.75000, -0.75000) -- (0.00000, 0.00000, 1.00000);

\node[vertex] at (-0.75000, -0.75000, -0.75000)     {};
\node[vertex] at (0.75000, -0.75000, -0.75000)     {};
\node[vertex] at (-0.75000, 0.75000, -0.75000)     {};
\node[vertex] at (0.75000, 0.75000, -0.75000)     {};
\node[vertex] at (0.00000, 0.00000, 1.00000)     {};

\end{tikzpicture}

\end{document}

Answer 1

It is rather easy to create something similar with asymptote.

 \documentclass{article}
 \usepackage[inline]{asymptote}
 \begin{document}
 \thispagestyle{empty}
 % from http://www.piprime.fr/files/asymptote/modules/polyhedron_js/index.html
 \begin{asy}
import polyhedron_js;
import three;

currentprojection=perspective(6,6,-1);

// comment the following line for OpenGl
// settings.render=0;

settings.tex="pdflatex";
settings.outformat="pdf"; // for opacity

// uncomment these lines for the coordinate axes
//   real r=1.5;
//   draw(Label("$x$",1), O--r*X, Arrow3(HookHead3));
//   draw(Label("$y$",1), O--r*Y, Arrow3(HookHead3));
//   draw(Label("$z$",1), O--r*Z, Arrow3(HookHead3));

polyhedron pyramid;
real sq2=sqrt(2)/2; real sq4=sqrt(2)/4;
pyramid[0]=(-sq2,-sq2,0)--(-sq2,sq2,0)--(sq2,sq2,0)--(sq2,-sq2,0)--cycle;
pyramid[1]=(sq2,-sq2,0)--(sq2,sq2,0)--(0,0,1)--cycle;
pyramid[2]=(sq2,sq2,0)--(-sq2,sq2,0)--(0,0,1)--cycle;
pyramid[3]=(-sq2,sq2,0)--(-sq2,-sq2,0)--(0,0,1)--cycle;
pyramid[4]=(-sq2,-sq2,0)--(sq2,-sq2,0)--(0,0,1)--cycle;
size(10cm);
polyhedron[] parr={shift(0,0,-sq2/2)*rotate(0,Z)*pyramid};
filldraw(parr,new pen[]{0.9blue},op=1); //

//  if(!is3D())
//    shipout(bbox(3mm,darkblue+3bp+miterjoin,FillDraw(paleblue))); 

//    draw(shift(0,0,0.5)*scale3(1)*surface(unitcircle3),opacity(0.2));

real radius=1.8;
path3 p3=O--arc(O,radius,0,0,180,0)--cycle; // define a wedge
draw(shift(0sq2/2,0sq2/2,0)*rotate(45,X)*surface(p3), blue+opacity(0.2));   
draw(shift(0sq2/2,0sq2/2,0)*rotate(-45,Y)*rotate(90,Z)*surface(p3), blue+opacity(0.2)); 
path3 p4=O--arc(O,radius,0,0,90,0)--cycle; // define a wedge
draw(shift(0sq2/2,0sq2/2,0)*rotate(90,Z)*rotate(90,X)*surface(p4), blue+opacity(0.2));  
// draw(shift(0sq2/2,0sq2/2,0)*rotate(45,Z)*rotate(45,X+Y)*rotate(45,Z)*surface(p4), blue+opacity(0.2));    
\end{asy}
\end{document}

You need to run xelatex, then there will be an .asy file, which you need to process with asy, and then you need to run xelatex twice. This is just a starting point, I did not make too much of an effort because I did not know whether you consider asymptote based solutions at all.