使用 tikz 或其他任何工具绘制三阶矩阵行列式的演示图

Question

我尝试使用 TikZ 和 绘制下面的图片\tikzmarknode。

直线段很容易画，但是曲线部分怎么画呢？

我的代码：

\documentclass{article}
\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{tikzmark}
\begin{document}

\[\begin{vmatrix}
  \tikzmarknode{11}{a_{11}} & \tikzmarknode{12}{a_{12}} & \tikzmarknode{13}{a_{13}} \\
  \tikzmarknode{21}{a_{21}} & \tikzmarknode{22}{a_{22}} & \tikzmarknode{23}{a_{23}} \\
  \tikzmarknode{31}{a_{31}} & \tikzmarknode{32}{a_{32}} & \tikzmarknode{33}{a_{33}}
\end{vmatrix}
\tikz[remember picture, overlay]\draw (11)--(22)--(33);
\tikz[remember picture, overlay]\draw (12)--(23);
\tikz[remember picture, overlay]\draw (21)--(32);
\tikz[remember picture, overlay]\draw[densely dashed] (31)--(22)--(13);
\tikz[remember picture, overlay]\draw[densely dashed] (21)--(12);
\tikz[remember picture, overlay]\draw[densely dashed] (32)--(23);\]

\end{document}

Answer 1

某些东西与此相关（这些值是通过反复试验找到的近似值）：

\documentclass{article}
\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{tikzmark}

\begin{document}

\[\begin{vmatrix}
  \tikzmarknode{11}{a_{11}} & \tikzmarknode{12}{a_{12}} & \tikzmarknode{13}{a_{13}} \\
  \tikzmarknode{21}{a_{21}} & \tikzmarknode{22}{a_{22}} & \tikzmarknode{23}{a_{23}} \\
  \tikzmarknode{31}{a_{31}} & \tikzmarknode{32}{a_{32}} & \tikzmarknode{33}{a_{33}}
\end{vmatrix}\]

\begin{tikzpicture}[remember picture, overlay]
    \draw (11)--(22)--(33);
    \draw (12)--(23);
    \draw (21)--(32);
    \draw[densely dashed] (31)--(22)--(13);
    \draw[densely dashed] (21)--(12);
    \draw[densely dashed] (32)--(23);

    \draw[rounded corners] (31)--++(1.25,-0.65)--++(1.25,0.65)--(23);
    \draw[rounded corners] (32)--++(1.25,-0.65)--++(1.25,0.65)--(13);
    \draw[rounded corners, densely dashed] (11)--++(1.25,0.65)--++(1.25,-0.65)--(23);
    \draw[rounded corners, densely dashed] (12)--++(1.25,0.65)--++(1.25,-0.65)--(33);
\end{tikzpicture}

\end{document}

请注意，由于使用了overlay，线条可能会覆盖矩阵前后的文本。因此，您可能需要添加一些额外的间距。

---

另一种方法是使用包vNiceMatrix提供的nicematrix。然后您可以直接引用矩阵节点：

\documentclass{article}
\usepackage{nicematrix, tikz}

\begin{document}

\[\begin{vNiceMatrix}
    a_{11} & a_{12} & a_{13} \\
    a_{21} & a_{22} & a_{23} \\
    a_{31} & a_{32} & a_{33}
\CodeAfter
    \begin{tikzpicture}
        \draw (1-1)--(2-2)--(3-3);
        \draw (1-2)--(2-3);
        \draw (2-1)--(3-2);
        \draw[densely dashed] (3-1)--(2-2)--(1-3);
        \draw[densely dashed] (2-1)--(1-2);
        \draw[densely dashed] (3-2)--(2-3);
        
        \draw[rounded corners] (3-1)--++(1.25,-0.65)--++(1.25,0.65)--(2-3);
        \draw[rounded corners] (3-2)--++(1.25,-0.65)--++(1.25,0.65)--(1-3);
        \draw[rounded corners, densely dashed] (1-1)--++(1.25,0.65)--++(1.25,-0.65)--(2-3);
        \draw[rounded corners, densely dashed] (1-2)--++(1.25,0.65)--++(1.25,-0.65)--(3-3);
    \end{tikzpicture}
\end{vNiceMatrix}\]

\end{document}

Answer 2

TikZ-CD 的替代方案。

该vmatrix风格不会产生完全相同的矩阵……但是足够接近吗？

代码

\documentclass{article}
\usepackage{tikz-cd,amsmath}
\usetikzlibrary{ext.paths.ortho}
\tikzcdset{
  no arrow tips/.style={arrows=-},
  math matrix/.style={
    column sep=2*\the\arraycolsep, row sep=+0pt,
    cells={nodes={inner xsep=+0pt, outer sep=.1ex, inner ysep=.26\baselineskip}}},
  vmatrix/.style={math matrix, /tikz/every matrix/.append style={
    outer xsep=+-.14em, left delimiter=|, right delimiter=|}}}
\tikzcdset{
  --/.style={dash pattern=on1.5ptoff1.5pt},
  DU/.style={DU path={#1}, rotate=-45},
  UD/.style={DU path={#1}, rotate=45}}
\tikzset{
  DU path/.default=1, DU path/.style={rounded corners,
    to path={r-rl[rl distance=#1*.5cm](\tikztotarget)}}}
\begin{document}
\[
\begin{tikzcd}[vmatrix, no arrow tips]
  a_{11} \drar           & a_{12} \drar           & a_{13} \\
  a_{21} \drar \urar[--] & a_{22} \drar \urar[--] & a_{23} \\
  a_{31}       \urar[--] & a_{32}       \urar[--] & a_{33}
  \arrow[from=3-1, to=2-3, DU]
  \arrow[from=3-2, to=1-3, DU=2]
  \arrow[from=3-3, to=1-2, UD=2, --]
  \arrow[from=2-3, to=1-1, UD,   --]
\end{tikzcd}
\]
\[
\begin{tikzcd}[vmatrix, no arrow tips,
  DU/.style={DU path={#1}, rotate=-45, yslant=.3333},
  UD/.style={DU path={#1}, rotate= 45, yslant=-.3333}
] % only changes are the DU and UD paths
  a_{11} \drar           & a_{12} \drar           & a_{13} \\
  a_{21} \drar \urar[--] & a_{22} \drar \urar[--] & a_{23} \\
  a_{31}       \urar[--] & a_{32}       \urar[--] & a_{33}
  \arrow[from=3-1, to=2-3, DU]
  \arrow[from=3-2, to=1-3, DU=2]
  \arrow[from=3-3, to=1-2, UD=2, --]
  \arrow[from=2-3, to=1-1, UD,   --]
\end{tikzcd}
\]
\end{document}

输出