TikZ - Spherical triangles

Question

I would like to draw this picture:

The triangle ABC is a spherical one.

How can I do this using TikZ?

Answer 1

Here is an example made in 3d. I put in it some parameters, but it's not so very flexible. That was to avoid too long trigonometric calculations. The circles can be drawn using the canvas and rotate around options, and clipping them.

This is the code:

\documentclass[border=2mm]{standalone}
\usepackage    {tikz}
\usetikzlibrary{3d}
\usetikzlibrary{calc}
\def\xx{0.5} % reduction x axis, cavalier perspective
\def\aa{30}  % angles AOB=BOC
\def\r {3}   % radius
\def\l {4.5} % distance AD=AE
\pgfmathsetmacro\ay{\rcos(\aa)} % Coordinates A
\pgfmathsetmacro\az{\rsin(\aa)} 
\pgfmathsetmacro\cx{\rsin(\aa)} % Coordinates C
\pgfmathsetmacro\cy{\rcos(\aa)} 
\pgfmathsetmacro\nx{-sin(\aa)cos(\aa)} % normal vector OAC plane
\pgfmathsetmacro\ny{ sin(\aa)sin(\aa)}
\pgfmathsetmacro\nn{sqrt(2\nx\nx+\ny*\ny)} % modulus
\pgfmathsetmacro\ap{acos(abs(\nx)/\nn)} % angle between planes XY and OAC
\begin{document}
\begin{tikzpicture}[line cap=round,line join=round,x={(-\xx cm,-\xx cm)},y={(1 cm,0 cm)},z={(0 cm,1 cm)}]
  % coordinates
  \coordinate (O) at (0,0,0);
  \coordinate (A) at (0,\ay,\az);
  \coordinate (B) at (0,\r,0);
  \coordinate (C) at (\cx,\cy,0);
  \coordinate (D) at (0,\l,0);
  \coordinate (E) at ({\lsin(\aa)},{\lcos(\aa)},0);
  % labels
  \node      at (O) [left]  {$O$};
  \node      at (A) [above] {$A$};
  \node      at (B) [below] {$B$};
  \node      at (C) [below] {$C$};
  \node      at (D) [below] {$D$};
  \node      at (E) [below] {$E$};
  \node[red] at ($(B)!0.5!(C)$) [below] {$a$};
  \node[red] at ($(A)!0.5!(C)$)         {$b$};
  \node[red] at ($(A)!0.5!(B)$) [right] {$c$};
  % spheric triangle
  \begin{scope} [canvas is xy plane at z=0]
    \clip (O) -- (D) -- (E) -- cycle;
    \draw[red] (O) circle (\r);
  \end{scope}
  \begin{scope}[canvas is yz plane at x=0]
    \clip (O) -- (A) -- (D) -- cycle;
    \draw[red] (O) circle (\r);
  \end{scope}
  \begin{scope}[rotate around z=-\aa, rotate around y=\ap-90,canvas is yz plane at x=0]
    \clip (O) -- (A) -- (E) -- cycle;
    \draw[red] (O) circle (\r);
  \end{scope}
  % lines
  \draw (O) -- (A) -- (D) -- (E) -- cycle;
  \draw[dashed] (O) -- (D);
  \draw (A) -- (E);
\end{tikzpicture}
\end{document}

And this is the triangle:

Answer 2

You can use tikz-3dplot.

\documentclass[tikz,border=3mm]{standalone}
\usepackage{tikz-3dplot}
\begin{document}
\tdplotsetmaincoords{70}{35}
\begin{tikzpicture}[tdplot_main_coords,line cap=round,line join=round,
    declare function={Px=0.2;Py=0.4;Pz=0.6;}]
    \draw (0,0,0) coordinate[label=left:{$O$}] (O) 
    (2,0,0) coordinate[label=below:{$A$}] (A) edge (O)
    -- (2,2,0) coordinate[label=below:{$B$}] (B) edge (O)
    -- (2,1,2) coordinate[label=above:{$C$}] (C) edge (O)
    -- (A)
    -- (3,0,0) coordinate[label=below:{$E$}] (E) edge (C)
    -- (3,3,0) coordinate[label=right:{$D$}] (D) edge (C)
    -- (B);
\end{tikzpicture}   
\end{document}