TikZ - 球面三角形

Question

我想画这样一幅画：

三角形ABC是球面三角形.

我如何使用 TikZ 做到这一点？

Answer 1

这是用 3d 制作的一个例子。我在其中加入了一些参数，但它不太灵活。这是为了避免太长的三角函数计算。可以使用canvas和rotate around选项绘制圆，然后剪切它们。

这是代码：

\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{\r*cos(\aa)} % Coordinates A
\pgfmathsetmacro\az{\r*sin(\aa)} 
\pgfmathsetmacro\cx{\r*sin(\aa)} % Coordinates C
\pgfmathsetmacro\cy{\r*cos(\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 ({\l*sin(\aa)},{\l*cos(\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}

这是三角形：

Answer 2

您可以使用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}