Tikz：定义点/极坐标点、路径、交点

下面一个例子展示如何使用Tikz完成：

• 定义一个坐标点，以备后面画线时直接使用；
• 定义一个极坐标点，以备在圆上快速定点；
• 根据定义的点进行路径的定义；
• 根据定义的路径计算路径的交点；
• 根据定义的坐标点进行封闭区域/形状的旋转！

注意：以上的所有操作不需要额外导入Tikz中的其他库，直接导入\usepackages{tikz}就可以实现上面的各种功能。

% 极坐标定点示例：长度不带单位，默认坐标单位一样！ —— 方便！
% 极坐标更方便的一点：代码改动小！只用调角度即可！
\begin{figure}[H]
\centering
\begin{tikzpicture}[scale = 3]
    %\draw [thick, <->] (-2,2.5) node [left] {$x_2$} -- (-2,-2) -- (2.5,-2) node [below right] {$x_1$};  
    \draw (0,0) circle [radius = 1];  % 注意：为了普通坐标和极坐标可以通用，把圆心定在原点！
    % 第一个三角形：
    \coordinate (O) at (0:0);  % 原点
    \coordinate (A) at (90:1);  
    \coordinate (B) at (210:1);
    \coordinate (C) at (-30:1);
    \draw [red] (A) -- (B) -- (C) -- (A);
    % 第二个三角形：
    \coordinate (D) at (-90:1);  
    \coordinate (E) at (30:1);
    \coordinate (F) at (150:1);
    \draw [green] (D) -- (E) -- (F) -- (D);
    % 定义路径：不画出来
    \path [name path = AB] (A) -- (B); \path [name path = AC] (A) -- (C); \path [name path = BC] (B) -- (C);
    \path [name path = DE] (D) -- (E); \path [name path = DF] (D) -- (F); \path [name path = EF] (E) -- (F);
    % 算交点：不画出来
    \path [ name intersections = {of = AB and EF, by = i1} ]; 
    \path [ name intersections = {of = AB and DF, by = i2} ];
    \path [ name intersections = {of = AC and EF, by = i3} ];
    \path [ name intersections = {of = AC and DE, by = i4} ];
    \path [ name intersections = {of = BC and DF, by = i5} ];
    \path [ name intersections = {of = BC and DE, by = i6} ];     
    % 填充：
    %\draw [very thin, fill = green, opacity = 0.5] (A) -- (i1) -- (i3) -- (A);
    \draw [very thin, fill = green, opacity = 0.5] (E) -- (i3) -- (i4) -- (E);
    %\draw [very thin, fill = green, opacity = 0.5] (C) -- (i4) -- (i6) -- (C);
    \draw [very thin, fill = green, opacity = 0.5] (D) -- (i5) -- (i6) -- (D);
    %\draw [very thin, fill = green, opacity = 0.5] (B) -- (i2) -- (i5) -- (B);
    \draw [very thin, fill = green, opacity = 0.5] (F) -- (i1) -- (i2) -- (F);
    
    % 旋转一个图形：在scope环境中
    \begin{scope}[rotate around={33:(O)}]
        % 坐标需要重定一遍
        \coordinate (A1) at (90:1);  
        \coordinate (B1) at (210:1);
        \coordinate (C1) at (-30:1);
        \draw[blue] (A1) -- (B1) -- (C1) -- cycle;  % cycle写最后表示闭环
    \end{scope}
\end{tikzpicture}
\caption{定义点+极坐标点、路径、交点、旋转}
\end{figure}

效果如下：