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tikz 中正 n 边形上的二面角作用

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tikz 中正 n 边形上的二面角作用

我试图在 tikz 中表示二面体群 (D_{2n}) 在正 n 边形上的作用，特别是 D_10 在五边形上的作用。我希望通过对节点进行标记和着色来表示它如何响应反射和旋转。

我对 tikz 还很陌生（在以前的工作中我只需要 tikz-cd），所以进展缓慢。

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{tikz}
\usetikzlibrary{shapes.geometric}
\begin{document}
    \begin{tikzpicture}[help lines/.style={blue!30,very thin}, thicc/.style={very thick}]
        \draw[help lines]   (-3.9,-3.9) grid (3.9,3.9);

        \node[thicc] (1) at (450:2) [circle,draw,red,fill=red!50,label=right:$1$];
        \node[thicc] (2) at (378:2) [circle,draw,blue,fill=blue!50,label=right:$2$];   
        \node[thicc] (3) at (306:2) [circle,draw,green,fill=green!30,label=right:$3$];    
        \node[thicc] (4) at (234:2) [circle,draw,orange,fill=red!30!yellow!30,label=right:$4$];
        \node[thicc] (5) at (162:2) [circle,draw,purple,fill=red!50!blue,label=right:$5$];

        \draw[thicc] (1) -- (2) -- (3) -- (4) -- (5) -- (1) -- cycle;
    \end{tikzpicture}

    \end{document}

我的目标是每个节点都有不同的颜色和标签（标签 1-5），然后重新绘制相同的形状，但反映在通过（比如说）顶点 2 的线上（可能用虚线表示对称线），即对应于排列 (13)(45) 和一些旋转，比如说 (13524)。

这是最好的方法吗？我尝试了 \foreach，但不确定如何正确使用语法来获取每个节点的不同颜色等。我还想将节点缩放得更大，并让标签出现在圆圈之外（而不是在某个固定方向上）。

我的做法正确吗？

答案1

为此，将五边形存储在 pic 中，然后对其进行变换可能更有意义。为了让标签始终直立，可能需要使用transform shape=false。将颜色存储在列表中并使用 foreach 循环可能也更方便。当然，相对于旋转轴的反射是一系列操作

rotate, reflect, inverse rotate

这是代码。

\documentclass[tikz,border=3.14mm]{standalone}
\usetikzlibrary{shapes.geometric}
\begin{document}
\begin{tikzpicture}[help lines/.style={blue!30,very thin},
pics/pentagon/.style={code={
\node[regular polygon,regular polygon sides=5,minimum size=4cm,draw]
        (5gon){};
 \foreach \X in {1,...,5}
  {\pgfmathsetmacro{\myfillcolor}{{\LstFillCols}[mod(\X-1,5)]}
  \pgfmathsetmacro{\mydrawcolor}{{\LstDrawCols}[mod(\X-1,5)]}
  \path (5gon.center) -- (5gon.corner \X) 
    node[circle,draw=\mydrawcolor,red,fill=\myfillcolor]{} 
    node[pos=1.2,transform shape=false](\X){\X};}}}]
 thicc/.style={very thick}]
 \edef\LstFillCols{"red!50","blue!50","green!30","red!30!yellow!30","red!50!blue"}
 \edef\LstDrawCols{"red","blue","green","orange","purple"}
 \draw[help lines]   (-3.9,-3.9) grid (3.9,3.9);
 \pic{pentagon};
 \begin{scope}[xshift=8cm]
  \draw[help lines]   (-3.9,-3.9) grid (3.9,3.9);
  \begin{scope}[rotate=72,xscale=-1,rotate=-72,transform shape]
   \draw[dashed] (162:3.9) -- (162+180:3.9); % symmetry axis
   \pic{pentagon};
  \end{scope}
 \end{scope}
 \end{tikzpicture}
\end{document}

在此处输入图片描述

这种方法的优点是它可以用于任意 n。

\documentclass[tikz,border=3.14mm]{standalone}
\usetikzlibrary{shapes.geometric}
\begin{document}
\begin{tikzpicture}[help lines/.style={blue!30,very thin},
 pics/ngon/.style={code={\tikzset{ngon/.cd,#1}
  \node[regular polygon,regular polygon sides=\pgfkeysvalueof{/tikz/ngon/n},
  minimum size=4cm,draw,ngon/border] (\pgfkeysvalueof{/tikz/ngon/name}){};
   \foreach \X in {1,...,\pgfkeysvalueof{/tikz/ngon/n}}
   {\pgfmathsetmacro{\myfillcolor}{{\LstFillCols}[mod(\X-1,5)]}
    \pgfmathsetmacro{\mydrawcolor}{{\LstDrawCols}[mod(\X-1,5)]}
   \path (\pgfkeysvalueof{/tikz/ngon/name}.center) -- (\pgfkeysvalueof{/tikz/ngon/name}.corner \X) 
    node[circle,draw=\mydrawcolor,fill=\myfillcolor,ngon/nodes]{} 
    node[pos=\pgfkeysvalueof{/tikz/ngon/label pos},transform shape=false](\X){\X};
    }
    }},flip about/.style={/utils/exec=\pgfmathsetmacro{\posangle}{%
    -1*iseven(\pgfkeysvalueof{/tikz/ngon/n})*180/\pgfkeysvalueof{/tikz/ngon/n}+%
    (#1-1)*360/\pgfkeysvalueof{/tikz/ngon/n}},
    rotate=\posangle,xscale=-1,rotate=-1*\posangle},
    ngon/.cd,n/.initial=5,border/.style={very thick},nodes/.style={very thick},
    name/.initial={ngon},label pos/.initial=1.2,
    angle of/.code 2 args=\pgfmathsetmacro{#2}{%
    -1*iseven(\pgfkeysvalueof{/tikz/ngon/n})*180/\pgfkeysvalueof{/tikz/ngon/n}+%
    (#1-1)*360/\pgfkeysvalueof{/tikz/ngon/n}}]
 \edef\LstFillCols{"red!50","blue!50","green!30","red!30!yellow!30","red!50!blue"}
 \edef\LstDrawCols{"red","blue","green","orange","purple"}
 \draw[help lines]   (-3.9,-3.9) grid (3.9,3.9);
 \pic{ngon={n=5}};
 \begin{scope}[xshift=8cm]
  \draw[help lines]   (-3.9,-3.9) grid (3.9,3.9);
  \begin{scope}[ngon/n=5,flip about=2,transform shape]
   \pgfkeys{tikz/ngon/angle of={2}{\myangle}}
   \draw[dashed] (\myangle+90:3.9) -- (\myangle+270:3.9); % symmetry axis
   \pic{ngon};
  \end{scope}
 \end{scope}
 % second example
 \begin{scope}[yshift=-8cm]
 \draw[help lines]   (-3.9,-3.9) grid (3.9,3.9);
 \pic{ngon={n=8}};
 \begin{scope}[xshift=8cm]
  \draw[help lines]   (-3.9,-3.9) grid (3.9,3.9);
  \begin{scope}[ngon/n=10,flip about=3,transform shape]
   \pgfkeys{tikz/ngon/angle of={3}{\myangle}}   
   \draw[dashed] (\myangle+90:3.9) -- (\myangle+270:3.9); % symmetry axis
   \pic{ngon};
  \end{scope}
 \end{scope}
 \end{scope}
 % third example
 % second example
 \begin{scope}[yshift=-16cm,ngon/.cd,n=5,
 nodes/.style={inner sep=2ex},label pos=1.4]
 \draw[help lines]   (-3.9,-3.9) grid (3.9,3.9);
 \pgfkeys{tikz/ngon/angle of={1}{\myangleA}}
 \pgfkeys{tikz/ngon/angle of={2}{\myangleB}}  
 \pic{ngon};
 \draw[semithick,-latex] (90+\myangleA:3.5) 
 arc(90+\myangleA:90+\myangleB:3.5);
 \begin{scope}[xshift=8cm]
  \draw[help lines]   (-3.9,-3.9) grid (3.9,3.9);
  \begin{scope}[flip about=2,transform shape]
   \pgfkeys{tikz/ngon/angle of={2}{\myangle}}
   \draw[dashed] (\myangle+90:3.9) -- (\myangle+270:3.9); % symmetry axis
   \pic{ngon};
  \end{scope}
 \end{scope}
 \end{scope}
 \end{tikzpicture}
\end{document}

在此处输入图片描述

在此代码中，有一个代码ngon/n可确定正多边形的顶点数。（还有其他键可确定线条、节点和名称的样式。）还有一个操作，它flip about=m就是上面提到的旋转、反射和逆旋转序列，它围绕穿过其第 1 个m角和中心的轴反射多边形。为方便起见，还有一个键ngon/angle of可计算第 1 个角的角度m（Ti钾Z 具有不同的约定，取决于角的数量是偶数还是奇数）。

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