LaTeX Tikz 绘图积累

Tikz 绘图的内容非常复杂，目前没有精力去仔细学习，只是简单记录一些可能用到的例子。一部分例子是我自己绘制的，更多的则是参考的网上公开资料。

Tikz 相关教程例如：TikZBolg。

有一些在线绘图工具可以可视化绘制简单图形，并导出 Tikz 代码，例如：Mathcha。

对于更复杂的需求，可以使用专业的矢量图绘制软件（例如 Inkscape） 绘制并导出 Tikz 代码。

函数图像

\documentclass[border=2mm]{standalone}
\usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage[dvipsnames]{xcolor}
\usepackage{tikz}

\begin{document}

\begin{tikzpicture}
    \draw[->] (-0.5, 0) -- (6, 0) node[above] {$x$};
    \draw[->] (0, -0.5) -- (0, 5) node[left] {$y$};

    \draw (2.5, 0.1) -- (2.5, -0.1) node[below] {$1$};
    \draw (-0.2, 0) node[below] {$0$};

    \draw[domain=0:1.0, thick, dashed, smooth, variable=\x] plot ({2.5*\x}, {5*\x*\x}) node[left] {$5x^2$};
    \draw[domain=0:2.2, thick, dashed, smooth, variable=\x] plot ({2.5*\x}, {\x*\x}) node[right] {$x^2$};
    \draw[domain=0:1.2, thick, smooth, variable=\x] plot ({2.5*\x}, {\x*\x + 2*\x + 1}) node[right] {$x^2 + 2x + 1$};
\end{tikzpicture}

\end{document}

\documentclass[border=2mm]{standalone}
\usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage[dvipsnames]{xcolor}
\usepackage{tikz}
\usetikzlibrary{calc}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}

\begin{document}

\begin{tikzpicture}
    \begin{axis}[
            xlabel=$n$ (Input Size),
            ylabel=$T(n)$ (Time Complexity),
            xmin=2, xmax=8,
            ymode=log,
            xtick={1, 2, 3, 4, 5, 6, 7, 8},
            ytick={1, 8, 64, 512, 4096},
            yticklabels={1, 8, 64, 512, 4096},
            domain=2:8,
            samples=200,
            clip=false,
        ]

        % O(1) - Constant
        \addplot[very thick, dashed] {1} node[right] {$O(1)$};

        % O(log n) - Logarithmic
        \addplot[very thick] {ln(x)} node[right] {$O(\log n)$};

        % O(n) - Linear
        \addplot[very thick] {x} node[right] {$O(n)$};

        % O(n log n) - Linearithmic
        \addplot[very thick] {x * ln(x)} node[right] {$O(n \log n)$};

        % O(n^2) - Quadratic
        \addplot[very thick] {x^2} node[right] {$O(n^2)$};

        % O(2^n) - Exponential
        \addplot[very thick] {2^x} node[right] {$O(2^n)$};

        \addplot[very thick,domain=2:7] {sqrt(2*pi*x) * (x/e)^x} node[above] {$O(n!)$}; % ~ n!

    \end{axis}
\end{tikzpicture}

\end{document}

\documentclass[border=2mm]{standalone}
\usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage[dvipsnames]{xcolor}
\usepackage{tikz}

\begin{document}

\begin{tikzpicture}
    \draw[->] (-2,0) -- (3,0) node[right] {$\mathbb{R}$};
    \draw[->] (0,-1) -- (0,2) node[above] {$\mathrm{i}\,\mathbb{R}$};
    \draw[->,thick] (0,0) -- (2,1) node[above right] {$z = a + b\,\mathrm{i}$};
    \draw[dashed] (2,1) -- (2,0);
    \draw[dashed] (2,1) -- (0,1);
    \draw (0.5,0) arc[start angle=0,end angle=26.57,radius=0.5];
    \node at (1,0.25) {$\theta$};
    \node at (1,.75) {$r$};
    \node at (2,-0.5) {$a = r \cos(\theta)$};
    \node at (-1.4,1) {$b = r \sin(\theta)$};
\end{tikzpicture}

\end{document}

\documentclass[border=2mm]{standalone}
\usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage[dvipsnames]{xcolor}
\usepackage{tikz}

\begin{document}

\begin{tikzpicture}
    \draw[domain=-4.5:4.5, smooth, variable=\x] plot ({\x}, {sin(\x r)}) node[right] {$\sin(x)$};
    \draw[domain=-4.5:4.5, smooth, variable=\x] plot ({\x}, {cos(\x r)}) node[right] {$\cos(x)$};

    \draw[->] (-5,0) -- (5,0) node [above] {$x$};
    \draw[->] (0, -1.2) -- (0, 1.2) node[left] {$y$};
\end{tikzpicture}

\end{document}

\documentclass[border=2mm]{standalone}
\usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage[dvipsnames]{xcolor}
\usepackage{tikz}

\begin{document}

\begin{tikzpicture}
    \draw[->] (-1,0) -- (4.7,0) node[right] {$x$};
    \draw[->] (0,-2.5) -- (0,2.5) node[above] {$y$};
    \draw[thick,orange,domain=0:3.7,smooth,variable=\x] plot ({\x},{(\x - 2)*(\x - 2) - 2}) node[right] {$f(x)$};

    \filldraw[thick, orange] (2, -2) circle (2pt) node[below right] {Global Min};

    \draw[] (0.1,2) -- (-0.1,2) node[left] {$\max_{[a,b]} f(x)$};
    \draw[dashed] (2,-2) -- (-0.1,-2) node[left] {$\min_{[a,b]} f(x)$};
    \draw[] (0.1,-2) -- (-0.1,-2);

    \node[below left] at (0,0) {$a$};
    \draw[dashed] (3.7, 0.89) -- (3.7,0) node[below] {$b$};

    \draw[dashed] (4, -1) -- (-0.1,-1) node[left] {$L$};

    \draw[dashed] (3, -1) -- (3,0) node[above] {$c_1$};
    \draw[dashed] (1, -1) -- (1,0) node[above] {$c_2$};
\end{tikzpicture}

\end{document}

\documentclass[border=2mm]{standalone}
\usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage[dvipsnames]{xcolor}
\usepackage{tikz}
\usetikzlibrary{calc}

\begin{document}

\begin{tikzpicture}[scale=0.7, font=\Large]
    \draw[->] (0,0) -- (11,0) node[pos=1.05] {$i$};
    \draw[->] (0,0) -- (0,5) node[anchor=south, pos=1] {$\sigma_i$};

    \draw[dashed] (0,1) -- (11,1);
    \draw (0,1) -- (0.2,1);
    \node[left] at (0,1) {$\alpha$};
    \node[left] at (0,0) {$0$};

    \def\radius{3pt}

    \foreach \i in {1,...,10} {
            \pgfmathsetmacro{\val}{4*exp(-0.5*(\i-1))}
            \draw[black, thick] (\i,\val) circle (\radius);
            \pgfmathsetmacro{\shiftVal}{max(\val - 1, 0)}
            \fill[red] (\i,\shiftVal) circle (\radius);
            \draw[->, red]
            ($(\i,\val) + (0,-\radius)$) --
            ($(\i,\shiftVal) + (0,\radius)$);
        }

\end{tikzpicture}

\end{document}

\documentclass{article}
\usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage[dvipsnames]{xcolor}
\usepackage{tikz}
\usepackage{subcaption}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}

\pagestyle{empty}

\begin{document}

\begin{figure}[htbp]
    \centering
    \begin{subfigure}[b]{0.45\textwidth}
        \centering
        \begin{tikzpicture}[scale=1.3]
            \draw[->] (-1.75, 0) -- (1.75, 0) node[right] {$x$};
            \draw[->] (0, -1.25) -- (0, 1.25) node[above] {$y$};
            \draw[thick, orange, domain=-1.5:1.5, samples=300] plot (\x, {cos(deg(1 / \x))});
        \end{tikzpicture}
        \caption*{$\cos\left(\frac{1}{x}\right)$}
    \end{subfigure}
    \begin{subfigure}[b]{0.45\textwidth}
        \centering
        \begin{tikzpicture}[scale=2]
            \draw[->] (-1.2, 0) -- (1.2, 0) node[right] {$x$};
            \draw[->] (0, -0.3) -- (0, 1.25) node[above] {$y$};
            \draw[thick, orange, domain=-1.05:1.05, samples=200] plot (\x, {cos(deg(1 / \x)) * (\x * \x)});
        \end{tikzpicture}
        \caption*{$x^2 \cos\left(\frac{1}{x}\right)$}
    \end{subfigure}
    \begin{subfigure}[b]{0.47\textwidth}
        \centering
        \begin{tikzpicture}[scale=.8]
            \draw[->] (-3,0) -- (3,0) node[right] {$x$};
            \draw[->] (0,-3) -- (0,3) node[above] {$y$};

            \draw[thick,orange,domain=0.37:2.5,smooth,variable=\x] plot ({\x},{1/\x}) node[right] {$f(x)$};
            \draw[thick,orange,domain=-2.5:-0.37,smooth,variable=\x] plot ({\x},{1/\x});
        \end{tikzpicture}
    \end{subfigure}
    \begin{subfigure}[b]{0.47\textwidth}
        \centering
        \begin{tikzpicture}[scale=1]
            \draw[->] (-3,0) -- (3,0) node[right] {$x$};
            \draw[->] (0,-1.5) -- (0,1.5) node[above] {$y$};
            \draw[thick,orange] (0,1) -- (2.5,1) node[right] {$f(x)$};
            \draw[thick,orange] (-2.5,-1) -- (0,-1);
        \end{tikzpicture}
    \end{subfigure}
    \begin{subfigure}[b]{0.45\textwidth}
        \centering
        % All norms
        \begin{tikzpicture}[scale=0.7]
            \begin{axis}[axis lines=middle,xtick=\empty,ytick=\empty,axis equal,enlargelimits,xmax=1.2,ymax=1.2,xmin=-1.2,ymin=-1.2]
                %p=0.5
                \begin{scope}[very thick,dotted,orange,domain=0:pi,samples=50]
                    \addplot[] ({(cos(deg(x)))^(4},{ (sin(deg(x))^(4});
                    \addplot[] ({(cos(deg(x)))^(4},{ -(sin(deg(x))^(4});
                    \addplot[] ({-(cos(deg(x)))^(4},{ (sin(deg(x))^(4});
                    \addplot[] ({-(cos(deg(x)))^(4},{-(sin(deg(x))^(4});
                \end{scope}
                %p=1
                \addplot[blue,domain=0:pi] ({(cos(deg(x)))^2},{(sin(deg(x))^2});
                \addplot[blue,domain=0:pi] ({(cos(deg(x)))^2},{-(sin(deg(x))^2});
                \addplot[blue,domain=0:pi] ({-(cos(deg(x)))^2},{(sin(deg(x))^2});
                \addplot[blue,domain=0:pi] ({-(cos(deg(x)))^2},{-(sin(deg(x))^2});
                %p=2
                \addplot[red,domain=-pi:0] ({(cos(deg(x)))},{(sin(deg(x))});
                \addplot[red,domain=0:pi] ({(cos(deg(x)))},{(sin(deg(x))});
                %p=inf
                \draw[thick,dashdotted,gray] (axis cs:-1,-1) rectangle (axis cs:1,1);
            \end{axis}
        \end{tikzpicture}
    \end{subfigure}
    \begin{subfigure}[b]{0.45\textwidth}
        \centering
        \begin{tikzpicture}[scale=0.7]
            \begin{axis}[axis lines=middle,xtick=\empty,ytick=\empty,axis equal,enlargelimits,xmax=1.5,ymax=1.5,xmin=-1.5,ymin=-1.5]
                %p=2
                \addplot[red,domain=-pi:0] ({(cos(deg(x)))},{(sin(deg(x)))});
                \addplot[red,domain=0:pi] ({(cos(deg(x)))},{(sin(deg(x)))});
                %p=2-larger
                \addplot[red,domain=-pi:0] ({(sqrt(2)*cos(deg(x)))},{((sqrt(2)*sin(deg(x))))});
                \addplot[red,domain=0:pi] ({(sqrt(2)*cos(deg(x)))},{((sqrt(2)*sin(deg(x))))});
                %p=inf
                \draw[thick,dashdotted,gray] (axis cs:-1,-1) rectangle (axis cs:1,1);
            \end{axis}
        \end{tikzpicture}
    \end{subfigure}
    % \caption{XXX} % \label{fig:XXX}
\end{figure}

\end{document}

线性代数

\documentclass{article}
\usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage[dvipsnames]{xcolor}
\usepackage{tikz}
\usepackage{subcaption}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}

\pagestyle{empty}

\begin{document}

\begin{figure}[htbp]
    \centering
    \begin{subfigure}[b]{0.45\textwidth}
        \centering
        \begin{tikzpicture}[scale=1]
            % Axes
            \draw[->] (-2,0,0) -- (2,0,0) node[right] {$x_2$};
            \draw[->] (0,-2,0) -- (0,2,0) node[above] {$x_3$};
            \draw[->] (0,0,-2) -- (0,0,2) node[above] {$x_1$};

            % Span of v
            \foreach \t in {-1.2,1.2} {
                    \draw[thick] (0,0,0) -- (\t*1,\t*2,\t*1);
                }

            % Vector v
            \draw[thick,->] (0,0,0) -- (1,2,1) node[midway,right] {$\vec{v}$};
        \end{tikzpicture}
    \end{subfigure}
    \begin{subfigure}[b]{0.45\textwidth}
        \centering
        \begin{tikzpicture}[scale=1]
            % Axes
            \draw[->] (-2,0,0) -- (2,0,0) node[right] {$x_2$};
            \draw[->] (0,-2,0) -- (0,2,0) node[above] {$x_3$};
            \draw[->] (0,0,-2) -- (0,0,2) node[above] {$x_1$};

            % Span of v1 and v2
            \fill[cyan!50,opacity=0.3] (-1.8,-1.2,-1.7) -- (1,2,1) -- (2,1,1) -- (-1,-2,-1) -- cycle;
            \draw[thick,cyan!50,opacity=0.3] (-1.8,-1.2,-1.7) -- (1,2,1) -- (2,1,1) -- (-1,-2,-1) -- cycle;

            % Vectors v1 and v2
            \draw[thick,->] (0,0,0) -- (1,2,1) node[above] {$\vec{v_1}$};
            \draw[thick,->] (0,0,0) -- (2,1,1) node[right] {$\vec{v_2}$};
        \end{tikzpicture}
    \end{subfigure}
    \begin{subfigure}[b]{0.47\textwidth}
        \centering
        \begin{tikzpicture}[scale=2.5]
            % Axes
            \draw[->] (-0.8,0) -- (1.2,0) node[right] {$x_1$};
            \draw[->] (0,-0.2) -- (0,1.2) node[above] {$x_2$};

            % Original vectors
            \draw[thick,->] (0,0) -- (1,0) node[midway, below] {$\vec{u}$};
            \draw[thick,->] (0,0) -- (0,1) node[midway, left] {$\vec{v}$};
            \draw[thick,->] (0,0) -- (-0.6,1) node[midway, left] {T($\vec{v})$};

            % Shaded area
            \fill[cyan!50,opacity=0.3] (0,0) -- (1,0) -- (0.4,1) -- (-0.6,1) -- cycle;
            \draw[dashed] (1,0) -- (1,1) -- (0,1);
        \end{tikzpicture}
    \end{subfigure}
    \begin{subfigure}[b]{0.47\textwidth}
        \centering
        \begin{tikzpicture}[scale=2.5]
            % Axes
            \draw[->] (-0.8,0,0) -- (1.2,0,0) node[right] {$x_1$};
            \draw[->] (0,-0.2,0) -- (0,1.2,0) node[above] {$x_2$};
            \draw[->] (0,0,0.2) -- (0,0,-1.2) node[below right] {$x_3$};

            % Original vectors
            \draw[thick,->] (0,0,0) -- (1,0,0) node[midway, below] {$\vec{u}$};
            \draw[thick,->] (0,0,0) -- (-0.6,1,0) node[midway, left] {$\vec{v}$};
            \draw[thick,->] (0,0,0) -- (0,0,-1) node[midway, right] {$\vec{w}$};

            % Shaded area
            \fill[cyan!50,opacity=0.3] (0,0,0) -- (1,0,0) -- (0.4,1,0) -- (-0.6,1,0) -- cycle;
            \fill[cyan!50,opacity=0.3] (0,0,0) -- (1,0,0) -- (1,0,-1) -- (0,0,-1) -- cycle;
            \draw[dashed] (1,0,0) -- (1,0,-1) -- (0,0,-1) -- (-0.6,1,-1) -- (-0.6,1,0) -- (0.4,1,0) -- cycle;
            \draw[dashed] (1,0,-1) -- (0.4,1,-1) -- (-0.6,1,-1);
            \draw[dashed] (0.4,1,0) -- (0.4,1,-1);
        \end{tikzpicture}
    \end{subfigure}
    \begin{subfigure}[b]{0.45\textwidth}
        \centering
        \begin{tikzpicture}[scale=2.5]
            % Axes
            \draw[->] (-0.2,0) -- (1.2,0) node[right] {$x_1$};
            \draw[->] (0,-0.2) -- (0,1.2) node[above] {$x_2$};

            \draw[thick,->] (0,0) -- (1,0) node[midway, below] {$\vec{u}$};
            \draw[thick,->] (0,0) -- (0,1) node[midway, left] {$\vec{v}$};

            \draw[thick,->] (0,0) -- (0.7,0.5) node[midway, above] {$\vec{w}$};

            \fill[cyan!50,opacity=0.3] (0,0) -- (1,0) -- (1,1) -- (0,1) -- cycle;
        \end{tikzpicture}
    \end{subfigure}
    \begin{subfigure}[b]{0.45\textwidth}
        \centering
        \begin{tikzpicture}[scale=2.5]
            % Axes
            \draw[->] (-0.2,0) -- (1.2,0) node[right] {$x_1$};
            \draw[->] (0,-0.2) -- (0,1.2) node[above] {$x_2$};

            \draw[thick,->] (0,0) -- (0.8,0.1) node[midway, below=0.1cm] {$T(\vec{u})$};
            \draw[thick,->] (0,0) -- (0.2,1.1) node[midway, left=0.2cm] {$T(\vec{v})$};

            \draw[thick,->] (0,0) -- (0.7,0.8) node[midway, above=0.3cm] {$T(\vec{w})$};

            \fill[cyan!50,opacity=0.3] (0,0) -- (0.8,0.1) -- (1,1.2) -- (0.2,1.1) -- cycle;
        \end{tikzpicture}
    \end{subfigure}
    \begin{subfigure}[b]{0.45\textwidth}
        \begin{tikzpicture}[scale=0.85]
            % Axes
            \draw[->] (0,0,0) -- (3,0,0) node[right] {$x_1$};
            \draw[->] (0,0,0) -- (0,2,0) node[above] {$x_3$};
            \draw[->] (0,0,0) -- (0,0,3) node[above] {$x_2$};

            % Projection onto x1-x2 plane
            \fill[cyan!50,opacity=0.3] (0,0,0) -- (3,0,0) -- (3,0,3) -- (0,0,3) -- cycle;

            % Point in space
            \fill[] (2,2,1) circle (2pt) node[above right] {$\vec{x}$};
            \fill[] (1,-1,2.5) circle (2pt) node[right] {$\vec{v}$};

            % Projection line
            \draw[dashed] (2,2,1) -- (2,0,1);
            \draw[dashed] (1,-1,2.5) -- (1,0,2.5);

            % Projection point
            \fill[orange] (2,0,1) circle (2pt) node[below right] {$T(\vec{x})$};
            \fill[orange] (1,0,2.5) circle (2pt) node[above right] {$T(\vec{v})$};
        \end{tikzpicture}
    \end{subfigure}
    % \caption{XXX} % \label{fig:XXX}
\end{figure}

\end{document}

几何图像

\documentclass[border=2mm]{standalone}
\usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage[dvipsnames]{xcolor}
\usepackage{tikz}

\begin{document}

\begin{tikzpicture}
    \coordinate (A) at (0,0);
    \coordinate (B) at (3,0);
    \coordinate (C) at (4,2);
    \coordinate (D) at (1,2);
    \draw[thick] (A) -- (B) -- (C) -- (D) -- cycle;
    \draw[->, thick, red] (A) -- (B) node[midway, below]{\(\vec{u}\)};
    \draw[->, thick, blue] (A) -- (D) node[midway, left]{\(\vec{v}\)};
    \foreach \p/\n in {A/A, B/B, C/C, D/D} {
            \filldraw (\p) circle (2pt) node[above right]{\n};
        }
\end{tikzpicture}

\end{document}

\documentclass[border=2mm]{standalone}
\usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage[dvipsnames]{xcolor}
\usepackage{tikz}
\usetikzlibrary{calc}

\begin{document}

\begin{tikzpicture}
    \coordinate (A) at (0,0);
    \coordinate (B) at (4,0);
    \coordinate (C) at (2,3);
    \draw[thick] (A) -- (B) -- (C) -- cycle;
    \draw[dashed, red] (C) -- ($(A)!(C)!(B)$) coordinate (H);
    \filldraw[black] (A) circle (2pt) node[below left]{A};
    \filldraw[black] (B) circle (2pt) node[below right]{B};
    \filldraw[black] (C) circle (2pt) node[above,yshift=2pt]{C};
    \filldraw[red] (H) circle (2pt) node[below,yshift=-2pt]{H};
\end{tikzpicture}

\end{document}

\documentclass[border=2mm]{standalone}
\usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage[dvipsnames]{xcolor}
\usepackage{tikz}
\usepackage{tkz-euclide}

\begin{document}

\begin{tikzpicture}[scale=1.5]
    \tkzDefPoints{0/0/A,5/0/B,5/5/F,0/5/E}
    \tkzDefPoints{2/2/D,7/2/C,7/7/G,2/7/H}
    \tkzDrawPolygon(A,B,F,E)
    \tkzDrawPolygon(G,C)
    \tkzDrawPolygon(G,H)
    \tkzDrawPolygon[dashed](A,D)
    \tkzDrawPolygon(H,E)
    \tkzDrawPolygon(G,F)
    \tkzDrawPolygon(B,C)
    \tkzDefMidPoint(A,C)
    \tkzDrawPolygon[dashed](D,C)
    \tkzDrawPolygon[dashed](D,H)
    \tkzGetPoint{I}
    \tkzDrawPoint(I)

    \tkzDefMidPoint(A,C)
    \tkzGetPoint{I}
    \tkzDrawPoint(I)

    \tkzDefMidPoint(A,H)
    \tkzGetPoint{N}
    \tkzDrawPoint(N)
    \tkzDefMidPoint(B,G)
    \tkzGetPoint{L}
    \tkzDrawPoint(L)

    \tkzDefMidPoint(H,F)
    \tkzGetPoint{J}
    \tkzDrawPoint(J)

    \tkzDefMidPoint(H,C)
    \tkzGetPoint{M}
    \tkzDrawPoint(M)

    \tkzDefMidPoint(A,F)
    \tkzGetPoint{K}
    \tkzDrawPoint(K)

    \tkzDrawPolygon(N,K,I,L,J,N,I)
    \tkzDrawPolygon(J,K,L)
    \tkzDrawPolygon[dashed](N,M)
    \tkzDrawPolygon[dashed](L,M)
    \tkzDrawPolygon[dashed](I,M)
    \tkzDrawPolygon[dashed](J,M)

    \tkzLabelPoints(A)
    \tkzLabelPoints(B)
    \tkzLabelPoints[below](I)
    \tkzLabelPoints[right](C,G,F)
    \tkzLabelPoints[left](D,H,E)

    \tkzLabelPoints[right](L)
    \tkzLabelPoints[left](N)
    \tkzLabelPoints[below](I)
    \tkzLabelPoints[above](J)
    \tkzLabelPoint[above left,xshift=5pt,yshift=3pt](K){$K$}
    \tkzLabelPoint[left,xshift=-1pt,yshift=3pt](M){$M$}
\end{tikzpicture}

\end{document}

\documentclass[border=2mm]{standalone}
\usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage[dvipsnames]{xcolor}
\usepackage{tikz}
\usepackage{tkz-euclide}

\begin{document}

\begin{tikzpicture}[scale=1.7]
    \coordinate (A) at (0,0);
    \coordinate (B) at (3,0);
    \coordinate (C) at (4,1);
    \coordinate (D) at (1,1);
    \coordinate (E) at (0,3);
    \coordinate (F) at (3,3);
    \coordinate (G) at (4,4);
    \coordinate (H) at (1,4);
    \coordinate (S) at (2,6);
    \coordinate (O) at (2,0.5);

    \draw (B) -- (C) -- (G) -- (F) -- (B) -- cycle;
    \draw[dashed] (G) -- (H) -- (E);
    \draw (E) -- (A) -- (B);
    \draw[dashed] (H) -- (D) -- (A);
    \draw[dashed] (D) -- (C);
    \draw (E) -- (S);
    \draw[dashed] (H) -- (S);
    \draw (G) -- (S);
    \draw (F) -- (S);
    \draw (E) -- (F);
    \draw[dashed] (A) -- (C);
    \draw[dashed] (B) -- (D);

    \node[below left] at (A) {A};
    \node[below right] at (B) {B};
    \node[right] at (C) {C};
    \node[above left,xshift=-2pt,yshift=-7pt] at (D) {D};
    \node[left,yshift=-3pt] at (E) {E};
    \node[above right,yshift=-11pt] at (F) {F};
    \node[right] at (G) {G};
    \node[left,xshift=15pt,yshift=-8pt] at (H) {H};
    \node[above right,xshift=-3pt] at (S) {S};
    \node[below left] at (O) {O};
\end{tikzpicture}

\end{document}

\documentclass[border=2mm]{standalone}
\usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage[dvipsnames]{xcolor}
\usepackage{tikz}
\usepackage{tkz-euclide}

\begin{document}

\begin{tikzpicture}[scale=1.7]
    \coordinate (A) at (0,0);
    \coordinate (B) at (3,0);
    \coordinate (C) at (1,2);
    \coordinate (D) at (0,4);
    \coordinate (E) at (3,4);
    \coordinate (F) at (1,6);
    \coordinate (S) at (7,3);
    \draw (A) -- (B);
    \draw (D) -- (E) -- (F) -- cycle;
    \draw (A) -- (D);
    \draw (B) -- (E);
    \draw[dashed] (F) -- (C) -- (A);
    \draw[dashed] (C) -- (B);
    \draw (F) -- (S);
    \draw (E) -- (S);
    \draw[dashed] (C) -- (S);
    \draw (B) -- (S);

    \node[xshift=-4pt,yshift=-7pt] at (A) {A};
    \node[xshift=4pt,yshift=-7pt] at (B) {B};
    \node[yshift=0pt,xshift=-8pt] at (C) {C};
    \node[xshift=-6pt] at (D) {D};
    \node[xshift=6pt,yshift=5pt] at (E) {E};
    \node[yshift=5pt] at (F) {F};
    \node[xshift=5pt] at (S) {S};
\end{tikzpicture}

\end{document}

一维空间离散网格

\documentclass[border=2mm]{standalone}
\usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage[dvipsnames]{xcolor}
\usepackage{tikz}

\begin{document}

\begin{tikzpicture}[>=stealth, thick]
    \def\N{6}
    \def\L{8}
    \def\H{0.15}

    \pgfmathsetmacro{\dx}{\L/\N}
    \pgfmathsetmacro{\halfdx}{0.5*\dx}

    % Axes
    \draw (0,0) -- (\L,0);

    % Points
    \foreach \i in {0,...,\N} {
            \pgfmathsetmacro{\x}{\i*\dx}
            \draw (\x,0) -- (\x,\H);
            \fill (\x,0) circle (0.4pt);
        }

    % Labels
    \draw[red] (0,0) -- (0,-\H);
    \node[below,red] at (0,-2*\H) {$a$};
    \draw[red] (\L,0) -- (\L,-\H);
    \node[below,red] at (\L,-2*\H) {$b$};

    \node[above] at (0,\H) {$x_{0}$};
    \node[above] at (\dx,\H) {$x_{1}$};
    \node[above] at (\L-\dx,\H) {$x_{N-1}$};
    \node[above] at (\L,\H) {$x_{N}$};
\end{tikzpicture}

\end{document}

\documentclass[border=2mm]{standalone}
\usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage[dvipsnames]{xcolor}
\usepackage{tikz}

\begin{document}

\begin{tikzpicture}[>=stealth, thick]
    \def\N{6}
    \def\L{8}
    \def\H{0.15}

    \pgfmathsetmacro{\dx}{\L/\N}
    \pgfmathsetmacro{\halfdx}{0.5*\dx}

    % Axes
    \draw[dashed,Green] (-\halfdx,0) -- (0,0);
    \draw[dashed,Green] (\L,0) -- (\L+\halfdx,0);
    \draw (0,0) -- (\L,0);

    % Points
    \foreach \i in {0,...,\N} {
            \pgfmathsetmacro{\x}{\i*\dx}
            \draw (\x,0) -- (\x,\H);
            \fill (\x,0) circle (0.4pt);
        }

    \pgfmathsetmacro{\BN}{\N + 1}
    \foreach \i in {0,...,\BN} {
            \pgfmathsetmacro{\x}{\i*\dx - 0.5*\dx}
            \draw[Green] (\x,0) -- (\x,-\H);
            \fill[Green] (\x,0) circle (0.4pt);
        }

    % Labels
    \draw[red] (0,0) -- (0,-\H);
    \node[below,red] at (0,-2*\H) {$a$};
    \draw[red] (\L,0) -- (\L,-\H);
    \node[below,red] at (\L,-2*\H) {$b$};

    \node[above] at (0,\H) {$x_{0}$};
    \node[above] at (\dx,\H) {$x_{1}$};
    \node[above] at (\L-\dx,\H) {$x_{N-1}$};
    \node[above] at (\L,\H) {$x_{N}$};

    \node[below,Green] at (-\halfdx,-\H) {$x_{-\frac{1}{2}}$};
    \node[below,Green] at (\halfdx,-\H) {$x_{\frac{1}{2}}$};
    \node[below,Green] at (\L-\halfdx,-\H) {$x_{N-\frac{1}{2}}$};
    \node[below,Green] at (\L+\halfdx,-\H) {$x_{N+\frac{1}{2}}$};
\end{tikzpicture}

\end{document}

\documentclass[border=2mm]{standalone}
\usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage[dvipsnames]{xcolor}
\usepackage{tikz}

\begin{document}

\begin{tikzpicture}[>=stealth, thick]
    \def\N{6}
    \def\L{8}
    \def\H{0.15}

    \pgfmathsetmacro{\dx}{\L/\N}
    \pgfmathsetmacro{\halfdx}{0.5*\dx}

    % Axes
    \draw[dashed,Green] (-\halfdx,0) -- (0,0);
    \draw[dashed,Green] (\L,0) -- (\L+\halfdx,0);
    \draw (0,0) -- (\L,0);

    % Points
    \foreach \i in {0,...,\N} {
            \pgfmathsetmacro{\x}{\i*\dx}
            \draw (\x,0) -- (\x,\H);
            \fill (\x,0) circle (0.4pt);
        }

    \pgfmathsetmacro{\BN}{\N + 1}
    \foreach \i in {0,...,\BN} {
            \pgfmathsetmacro{\x}{\i*\dx - 0.5*\dx}
            \draw[Green] (\x,0) -- (\x,-\H);
            \fill[Green] (\x,0) circle (0.4pt);
        }

    % Labels
    \draw[red] (0,0) -- (0,-\H);
    \node[below,red] at (0,-2*\H) {$a$};
    \draw[red] (\L,0) -- (\L,-\H);
    \node[below,red] at (\L,-2*\H) {$b$};

    \node[above] at (0,\H) {$x_{\frac12}$};
    \node[above] at (\dx,\H) {$x_{\frac32}$};
    \node[above] at (\L-\dx,\H) {$x_{N-\frac12}$};
    \node[above] at (\L,\H) {$x_{N+\frac12}$};

    \node[below,Green] at (-\halfdx,-\H) {$x_{0}$};
    \node[below,Green] at (\halfdx,-\H) {$x_1$};
    \node[below,Green] at (\L-\halfdx,-\H) {$x_{N}$};
    \node[below,Green] at (\L+\halfdx,-\H) {$x_{N+1}$};
\end{tikzpicture}

\end{document}

二维空间离散网格

下图应该是用其它专业软件制作后转换为 tikz 的，代码参考 https://arxiv.org/abs/2502.20883。

\documentclass[border=2mm]{standalone}
\usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage[dvipsnames]{xcolor}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
    %uncomment if require: \path (0,300);
    %set diagram left start at 0, and has height of 300

    %Shape: Grid [id:dp6086037867382692]
    \draw  [draw opacity=0][line width=1.5]  (286.78,55.22) -- (440.67,55.22) -- (440.67,198.45) -- (286.78,198.45) -- cycle ; \draw  [line width=1.5]  (304.51,55.22) -- (304.51,198.45)(360.51,55.22) -- (360.51,198.45)(416.51,55.22) -- (416.51,198.45) ; \draw  [line width=1.5]  (286.78,72.95) -- (440.67,72.95)(286.78,128.95) -- (440.67,128.95)(286.78,184.95) -- (440.67,184.95) ; \draw  [line width=1.5]   ;
    %Shape: Diamond [id:dp2704383770206539]
    \draw  [draw opacity=0][fill={rgb, 255:red, 10; green, 69; blue, 199 }  ,fill opacity=1 ] (304.51,151.7) -- (309.17,156.45) -- (304.51,161.2) -- (299.84,156.45) -- cycle ;
    %Shape: Diamond [id:dp05191167319843715]
    \draw  [draw opacity=0][fill={rgb, 255:red, 10; green, 69; blue, 199 }  ,fill opacity=1 ] (332.51,180.2) -- (337.17,184.95) -- (332.51,189.7) -- (327.84,184.95) -- cycle ;
    %Shape: Diamond [id:dp8396711776809587]
    \draw  [draw opacity=0][fill={rgb, 255:red, 10; green, 69; blue, 199 }  ,fill opacity=1 ] (360.51,152.2) -- (365.17,156.95) -- (360.51,161.7) -- (355.84,156.95) -- cycle ;
    %Shape: Diamond [id:dp43560726895372204]
    \draw  [draw opacity=0][fill={rgb, 255:red, 10; green, 69; blue, 199 }  ,fill opacity=1 ] (332.89,123.94) -- (337.55,128.69) -- (332.89,133.44) -- (328.22,128.69) -- cycle ;
    %Shape: Diamond [id:dp49146238603682535]
    \draw  [draw opacity=0][fill={rgb, 255:red, 10; green, 69; blue, 199 }  ,fill opacity=1 ] (388.51,180.2) -- (393.17,184.95) -- (388.51,189.7) -- (383.84,184.95) -- cycle ;
    %Shape: Diamond [id:dp625980383948383]
    \draw  [draw opacity=0][fill={rgb, 255:red, 10; green, 69; blue, 199 }  ,fill opacity=1 ] (416.51,152.2) -- (421.17,156.95) -- (416.51,161.7) -- (411.84,156.95) -- cycle ;
    %Shape: Diamond [id:dp6603613912995167]
    \draw  [draw opacity=0][fill={rgb, 255:red, 10; green, 69; blue, 199 }  ,fill opacity=1 ] (304.51,96.2) -- (309.17,100.95) -- (304.51,105.7) -- (299.84,100.95) -- cycle ;
    %Shape: Diamond [id:dp7532338499105224]
    \draw  [draw opacity=0][fill={rgb, 255:red, 10; green, 69; blue, 199 }  ,fill opacity=1 ] (360.51,96.86) -- (365.17,101.61) -- (360.51,106.36) -- (355.84,101.61) -- cycle ;
    %Shape: Diamond [id:dp08242433686903328]
    \draw  [draw opacity=0][fill={rgb, 255:red, 10; green, 69; blue, 199 }  ,fill opacity=1 ] (416.51,96.2) -- (421.17,100.95) -- (416.51,105.7) -- (411.84,100.95) -- cycle ;
    %Shape: Diamond [id:dp18820567957368406]
    \draw  [draw opacity=0][fill={rgb, 255:red, 10; green, 69; blue, 199 }  ,fill opacity=1 ] (332.51,68.2) -- (337.17,72.95) -- (332.51,77.7) -- (327.84,72.95) -- cycle ;
    %Shape: Diamond [id:dp7617641125065957]
    \draw  [draw opacity=0][fill={rgb, 255:red, 10; green, 69; blue, 199 }  ,fill opacity=1 ] (388.51,68.2) -- (393.17,72.95) -- (388.51,77.7) -- (383.84,72.95) -- cycle ;
    %Shape: Ellipse [id:dp12162460285712395]
    \draw  [draw opacity=0][fill={rgb, 255:red, 242; green, 7; blue, 37 }  ,fill opacity=1 ] (299.6,185) .. controls (299.6,182.23) and (301.8,180) .. (304.51,180) .. controls (307.22,180) and (309.42,182.23) .. (309.42,185) .. controls (309.42,187.76) and (307.22,190) .. (304.51,190) .. controls (301.8,190) and (299.6,187.76) .. (299.6,185) -- cycle ;
    %Shape: Ellipse [id:dp6029546507689866]
    \draw  [draw opacity=0][fill={rgb, 255:red, 242; green, 7; blue, 37 }  ,fill opacity=1 ] (299.6,128.97) .. controls (299.6,126.21) and (301.8,123.97) .. (304.51,123.97) .. controls (307.22,123.97) and (309.42,126.21) .. (309.42,128.97) .. controls (309.42,131.73) and (307.22,133.97) .. (304.51,133.97) .. controls (301.8,133.97) and (299.6,131.73) .. (299.6,128.97) -- cycle ;
    %Shape: Ellipse [id:dp8090189307248269]
    \draw  [draw opacity=0][fill={rgb, 255:red, 242; green, 7; blue, 37 }  ,fill opacity=1 ] (299.6,72.95) .. controls (299.6,70.19) and (301.8,67.95) .. (304.51,67.95) .. controls (307.22,67.95) and (309.42,70.19) .. (309.42,72.95) .. controls (309.42,75.71) and (307.22,77.95) .. (304.51,77.95) .. controls (301.8,77.95) and (299.6,75.71) .. (299.6,72.95) -- cycle ;
    %Shape: Ellipse [id:dp23527142386059752]
    \draw  [draw opacity=0][fill={rgb, 255:red, 242; green, 7; blue, 37 }  ,fill opacity=1 ] (355.62,72.95) .. controls (355.62,70.19) and (357.82,67.95) .. (360.53,67.95) .. controls (363.24,67.95) and (365.44,70.19) .. (365.44,72.95) .. controls (365.44,75.71) and (363.24,77.95) .. (360.53,77.95) .. controls (357.82,77.95) and (355.62,75.71) .. (355.62,72.95) -- cycle ;
    %Shape: Ellipse [id:dp4464329470756502]
    \draw  [draw opacity=0][fill={rgb, 255:red, 242; green, 7; blue, 37 }  ,fill opacity=1 ] (355.62,128.97) .. controls (355.62,126.21) and (357.82,123.97) .. (360.53,123.97) .. controls (363.24,123.97) and (365.44,126.21) .. (365.44,128.97) .. controls (365.44,131.73) and (363.24,133.97) .. (360.53,133.97) .. controls (357.82,133.97) and (355.62,131.73) .. (355.62,128.97) -- cycle ;
    %Shape: Ellipse [id:dp4381396312087227]
    \draw  [draw opacity=0][fill={rgb, 255:red, 242; green, 7; blue, 37 }  ,fill opacity=1 ] (355.6,184.95) .. controls (355.6,182.19) and (357.8,179.95) .. (360.51,179.95) .. controls (363.22,179.95) and (365.42,182.19) .. (365.42,184.95) .. controls (365.42,187.71) and (363.22,189.95) .. (360.51,189.95) .. controls (357.8,189.95) and (355.6,187.71) .. (355.6,184.95) -- cycle ;
    %Shape: Ellipse [id:dp37631485662632347]
    \draw  [draw opacity=0][fill={rgb, 255:red, 242; green, 7; blue, 37 }  ,fill opacity=1 ] (411.65,185) .. controls (411.65,182.23) and (413.84,180) .. (416.56,180) .. controls (419.27,180) and (421.46,182.23) .. (421.46,185) .. controls (421.46,187.76) and (419.27,190) .. (416.56,190) .. controls (413.84,190) and (411.65,187.76) .. (411.65,185) -- cycle ;
    %Shape: Ellipse [id:dp9228054619778103]
    \draw  [draw opacity=0][fill={rgb, 255:red, 242; green, 7; blue, 37 }  ,fill opacity=1 ] (411.6,128.95) .. controls (411.6,126.19) and (413.8,123.95) .. (416.51,123.95) .. controls (419.22,123.95) and (421.42,126.19) .. (421.42,128.95) .. controls (421.42,131.71) and (419.22,133.95) .. (416.51,133.95) .. controls (413.8,133.95) and (411.6,131.71) .. (411.6,128.95) -- cycle ;
    %Shape: Ellipse [id:dp9761113886790156]
    \draw  [draw opacity=0][fill={rgb, 255:red, 242; green, 7; blue, 37 }  ,fill opacity=1 ] (411.6,72.95) .. controls (411.6,70.19) and (413.8,67.95) .. (416.51,67.95) .. controls (419.22,67.95) and (421.42,70.19) .. (421.42,72.95) .. controls (421.42,75.71) and (419.22,77.95) .. (416.51,77.95) .. controls (413.8,77.95) and (411.6,75.71) .. (411.6,72.95) -- cycle ;
    %Shape: Ellipse [id:dp24235284312952232]
    \draw  [draw opacity=0][fill={rgb, 255:red, 242; green, 7; blue, 37 }  ,fill opacity=1 ] (327.68,101.02) .. controls (327.68,98.26) and (329.88,96.02) .. (332.59,96.02) .. controls (335.3,96.02) and (337.5,98.26) .. (337.5,101.02) .. controls (337.5,103.78) and (335.3,106.02) .. (332.59,106.02) .. controls (329.88,106.02) and (327.68,103.78) .. (327.68,101.02) -- cycle ;
    %Shape: Ellipse [id:dp1826735098607173]
    \draw  [draw opacity=0][fill={rgb, 255:red, 242; green, 7; blue, 37 }  ,fill opacity=1 ] (383.6,100.95) .. controls (383.6,98.19) and (385.8,95.95) .. (388.51,95.95) .. controls (391.22,95.95) and (393.42,98.19) .. (393.42,100.95) .. controls (393.42,103.71) and (391.22,105.95) .. (388.51,105.95) .. controls (385.8,105.95) and (383.6,103.71) .. (383.6,100.95) -- cycle ;
    %Shape: Ellipse [id:dp255823116982711]
    \draw  [draw opacity=0][fill={rgb, 255:red, 242; green, 7; blue, 37 }  ,fill opacity=1 ] (327.6,156.95) .. controls (327.6,154.19) and (329.8,151.95) .. (332.51,151.95) .. controls (335.22,151.95) and (337.42,154.19) .. (337.42,156.95) .. controls (337.42,159.71) and (335.22,161.95) .. (332.51,161.95) .. controls (329.8,161.95) and (327.6,159.71) .. (327.6,156.95) -- cycle ;
    %Shape: Ellipse [id:dp9494151404914138]
    \draw  [draw opacity=0][fill={rgb, 255:red, 242; green, 7; blue, 37 }  ,fill opacity=1 ] (383.6,156.95) .. controls (383.6,154.19) and (385.8,151.95) .. (388.51,151.95) .. controls (391.22,151.95) and (393.42,154.19) .. (393.42,156.95) .. controls (393.42,159.71) and (391.22,161.95) .. (388.51,161.95) .. controls (385.8,161.95) and (383.6,159.71) .. (383.6,156.95) -- cycle ;
    %Shape: Diamond [id:dp9243702003505755]
    \draw  [draw opacity=0][fill={rgb, 255:red, 10; green, 69; blue, 199 }  ,fill opacity=1 ] (388.51,124.2) -- (393.17,128.95) -- (388.51,133.7) -- (383.84,128.95) -- cycle ;

    % Text Node
    \draw (326.16,191) node [anchor=north west][inner sep=0.75pt]  [font=\footnotesize]  {$x_{i}$};
    % Text Node
    \draw (377.4,191) node [anchor=north west][inner sep=0.75pt]  [font=\footnotesize]  {$x_{i+1}$};
    % Text Node
    \draw (285,152) node [anchor=north west][inner sep=0.75pt]  [font=\footnotesize]  {$y_{j}$};
    % Text Node
    \draw (272,96) node [anchor=north west][inner sep=0.75pt]  [font=\footnotesize]  {$y_{j+1}$};
    % Text Node
    \draw (290,199) node [anchor=north west][inner sep=0.75pt]  [font=\footnotesize]  {$x_{i-\frac12}$};
    % Text Node
    \draw (348,199) node [anchor=north west][inner sep=0.75pt]  [font=\footnotesize]  {$x_{i+\frac12}$};
    % Text Node
    \draw (403,199) node [anchor=north west][inner sep=0.75pt]  [font=\footnotesize]  {$x_{i+\frac32}$};
    % Text Node
    \draw (250,180) node [anchor=north west][inner sep=0.75pt]  [font=\footnotesize]  {$y_{j-\frac12}$};
    % Text Node
    \draw (250,124) node [anchor=north west][inner sep=0.75pt]  [font=\footnotesize]  {$y_{j+\frac12}$};
    % Text Node
    \draw (250,68) node [anchor=north west][inner sep=0.75pt]  [font=\footnotesize]  {$y_{j+\frac32}$};
\end{tikzpicture}
\end{document}

tikz 批注公式

可以利用 tikz 宏包在给数学公式加上背景颜色以及标注。（受到 synercys/annotated_latex_equations 的启发）

实现原理比较复杂：

• 在公式环境中通过 \tikzmarknode 命令留下绘图锚点；
• 在 tikzpicture 环境中绘制标注，需要利用前面留下的绘图锚点；
• 使用 figure 环境包裹公式和标注作为浮动体（否则标注可能会与其他段落混杂在一起，难以保持合适间距）。

完整源码如下

\documentclass{article}
\usepackage{amsmath}
\usepackage[dvipsnames]{xcolor}
\usepackage{tikz}
\usetikzlibrary{tikzmark}

\pagestyle{empty}

\begin{document}


\begin{figure}[htbp]
    \centering
    \vspace{2\baselineskip}

    \begin{equation*}
        \rho
        \left(
        \frac{\partial \mathbf{u}}{\partial t}
        +
        \tikzmarknode{uconv}{\colorbox{Bittersweet!20}{$\mathbf{u}\cdot\nabla \mathbf{u}$}}
        \right)
        =
        -\tikzmarknode{gradp}{\colorbox{OliveGreen!20}{$\nabla p$}}
        +
        \tikzmarknode{visc}{\colorbox{Orange!40}{$\mu \nabla^2 \mathbf{u}$}}
        +
        \tikzmarknode{f}{\colorbox{RoyalBlue!20}{$\mathbf{f}$}}
    \end{equation*}

    \begin{tikzpicture}[overlay,remember picture,>=stealth,
            nodes={align=left,inner ysep=1pt},<-]
        % Convective term
        \path (uconv.south) ++ (0,-1.0em)
        node[anchor=north east,color=Bittersweet!85] (uconv_text)
        {\scriptsize convective acceleration};
        \draw[color=Bittersweet] (uconv.south) |- (uconv_text.north west);

        % Pressure gradient
        \path (gradp.north) ++ (0,1.0em)
        node[anchor=south east,color=OliveGreen!85] (gradp_text)
        {\scriptsize pressure gradient force};
        \draw[color=OliveGreen] (gradp.north) |- (gradp_text.south west);

        % Viscous diffusion
        \path (visc.south) ++ (0,-1.0em)
        node[anchor=north west,color=Orange!85] (visc_text)
        {\scriptsize viscous diffusion term};
        \draw[color=Orange] (visc.south) |- (visc_text.north east);

        % Body force
        \path (f.north) ++ (0,1.0em)
        node[anchor=south west,color=RoyalBlue!85] (f_text)
        {\scriptsize body force};
        \draw[color=RoyalBlue] (f.north) |- (f_text.south east);
    \end{tikzpicture}

    \vspace{1.5\baselineskip}
    \caption{Annotated Navier–Stokes momentum equation.}\label{fig:ns_annotated}
\end{figure}

\begin{figure}[htbp]
    \centering
    \vspace{2\baselineskip}

    \begin{equation*}
        \partial_t f + \tikzmarknode{transport}{\colorbox{Bittersweet!20}{$v \cdot \nabla_x f$}}
        + \tikzmarknode{acceleration}{\colorbox{OliveGreen!20}{$\frac{F}{m} \cdot \nabla_v f$}}
        =
        \tikzmarknode{collision}{\colorbox{RoyalBlue!20}{$C(f)$}}.
    \end{equation*}

    \begin{tikzpicture}[overlay,remember picture,>=stealth,
            nodes={align=left,inner ysep=1pt},<-]
        \path (transport.south) ++ (0,-1.0em)
        node[anchor=north east,color=Bittersweet!85] (transport_text)
        {\scriptsize transport};
        \draw[color=Bittersweet] (transport.south) |- (transport_text.north west);

        \path (acceleration.south) ++ (0,-1.0em)
        node[anchor=north west,color=OliveGreen!85] (acceleration_text)
        {\scriptsize acceleration};
        \draw[color=OliveGreen] (acceleration.south) |- (acceleration_text.north east);

        \path (collision.north) ++ (0,1.0em)
        node[anchor=south west,color=RoyalBlue!85] (collision_text)
        {\scriptsize collision};
        \draw[color=RoyalBlue] (collision.north) |- (collision_text.south east);
    \end{tikzpicture}

    \vspace{1.5\baselineskip}
    \caption{Annotated Boltzmann equation.}\label{fig:boltzmann_annotated}
\end{figure}

\begin{figure}[htbp]
    \centering
    \vspace{2\baselineskip}

    \begin{align*}
         & \partial_t f
        +
        \tikzmarknode{vp_transport}{\colorbox{Bittersweet!20}{$v\cdot\nabla_x f$}}
        +
        \tikzmarknode{vp_force}{\colorbox{OliveGreen!20}{$E\cdot\nabla_v f$}}
        =
        0,
        \\
         & E = -\nabla_x \phi,
        \quad
        -\Delta_x \phi = \int f\,dv - \rho_0.
    \end{align*}

    \begin{tikzpicture}[overlay,remember picture,>=stealth,
            nodes={align=left,inner ysep=1pt},<-]
        % Transport
        \path (vp_transport.north) ++ (0,1.0em)
        node[anchor=south east,color=Bittersweet!85] (vp_transport_text)
        {\scriptsize transport};
        \draw[color=Bittersweet] (vp_transport.north) |- (vp_transport_text.south west);

        % Force term
        \path (vp_force.north) ++ (0,1.0em)
        node[anchor=south west,color=OliveGreen!85] (vp_force_text)
        {\scriptsize acceleration};
        \draw[color=OliveGreen] (vp_force.north) |- (vp_force_text.south east);

    \end{tikzpicture}

    \vspace{1.5\baselineskip}
    \caption{Annotated Vlasov–Poisson equation.}
\end{figure}

\begin{figure}[htbp]
    \centering
    \vspace{2\baselineskip}

    \begin{equation*}
        \varepsilon\,
        \partial_t I
        +
        \tikzmarknode{rte_transport}{\colorbox{OliveGreen!20}{$\Omega\, \partial_x I$}}
        =
        \tikzmarknode{rte_relax}{\colorbox{RoyalBlue!20}{$\frac{\sigma^s}{\varepsilon}(\langle I\rangle - I)$}}
        -
        \tikzmarknode{rte_absorb}{\colorbox{Orange!40}{$\varepsilon\,\sigma^a I$}}
        +
        \tikzmarknode{rte_source}{\colorbox{YellowGreen!35}{$\varepsilon\,Q$}}
        \, .
    \end{equation*}

    \begin{tikzpicture}[overlay,remember picture,>=stealth,
            nodes={align=left,inner ysep=1pt},<-]
        % Transport
        \path (rte_transport.south) ++ (0,-1.0em)
        node[anchor=north east,color=OliveGreen!85] (rte_transport_text)
        {\scriptsize transport};
        \draw[color=OliveGreen] (rte_transport.south) |- (rte_transport_text.north west);

        % Collision relaxation
        \path (rte_relax.north) ++ (0,1.0em)
        node[anchor=south east,color=RoyalBlue!85] (rte_relax_text)
        {\scriptsize scattering};
        \draw[color=RoyalBlue] (rte_relax.north) |- (rte_relax_text.south west);

        % Absorption term
        \path (rte_absorb.south) ++ (0,-1.0em)
        node[anchor=north west,color=Orange!85] (rte_absorb_text)
        {\scriptsize absorption};
        \draw[color=Orange] (rte_absorb.south) |- (rte_absorb_text.north east);

        % Source term
        \path (rte_source.north) ++ (0,1.0em)
        node[anchor=south west,color=YellowGreen!95] (rte_source_text)
        {\scriptsize source term};
        \draw[color=YellowGreen] (rte_source.north) |- (rte_source_text.south east);

    \end{tikzpicture}

    \vspace{1.4\baselineskip}
    \caption{Annotated one-dimensional radiative transfer equation in diffusive scaling.}
\end{figure}


\begin{figure}[htbp]
    \centering
    \vspace{2\baselineskip}

    % Scaled TRT equation
    \begin{align*}
        \frac{\varepsilon}{c}\partial_t I
        +
        \tikzmarknode{transport}{\colorbox{OliveGreen!20}{$\Omega\cdot\nabla_x I$}}
        ={} &
        \tikzmarknode{relax}{\colorbox{RoyalBlue!20}{$\frac{\sigma^s}{\varepsilon} \left(\frac{\phi}{4\pi}-I\right)$}}
        -
        \tikzmarknode{abs}{\colorbox{Orange!40}{$\varepsilon\,\sigma^a I$}}
        +
        \tikzmarknode{emit}{\colorbox{YellowGreen!35}{$\varepsilon\,\sigma^a B(T)$}} .
        \\
        C_v\,\partial_t T
        ={} &
        \tikzmarknode{couple}{\colorbox{Plum!25}{$\sigma^a(\phi-4\pi B(T))$}} .
    \end{align*}

    \begin{tikzpicture}[overlay,remember picture,>=stealth,
            nodes={align=left,inner ysep=1pt},<-]
        % Transport
        \path (transport.north) ++ (0,1.0em)
        node[anchor=south east,color=Bittersweet!85] (transport_text)
        {\scriptsize transport};
        \draw[color=Bittersweet] (transport.north) |- (transport_text.south west);
        % relaxation
        \path (relax.north) ++ (0,1.0em)
        node[anchor=south west,color=RoyalBlue!85] (relax_text)
        {\scriptsize scattering};
        \draw[color=RoyalBlue] (relax.north) |- (relax_text.south east);

        % absorption
        \path (abs.north) ++ (0,1.0em)
        node[anchor=south west,color=Orange!85] (abs_text)
        {\scriptsize absorption};
        \draw[color=Orange] (abs.north) |- (abs_text.south east);

        % emission
        \path (emit.north) ++ (0,1.0em)
        node[anchor=south west,color=YellowGreen!85] (emit_text)
        {\scriptsize material thermal emission};
        \draw[color=YellowGreen] (emit.north) |- (emit_text.south east);

        % coupling
        \path (couple.south) ++ (0,-1.0em)
        node[anchor=north west,color=Plum!85] (couple_text)
        {\scriptsize radiation--material coupling};
        \draw[color=Plum] (couple.south) |- (couple_text.north east);
    \end{tikzpicture}

    \vspace{1.5\baselineskip}
    \caption{Scaled thermal radiative transfer equations in diffusive scaling.}
\end{figure}

\end{document}

tikz-cd 交换图

\documentclass[border=2mm]{standalone}
\usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage[dvipsnames]{xcolor}
\usepackage{tikz-cd}

\begin{document}

\begin{tikzcd}
    T
    \arrow[drr, bend left, "x"]
    \arrow[ddr, bend right, "y"]
    \arrow[dr, dotted, "{(x,y)}" description] & & \\
    & X \times_Z Y \arrow[r, "p"] \arrow[d, "q"]
    & X \arrow[d, "f"] \\
    & Y \arrow[r, "g"]
    & Z
\end{tikzcd}

\end{document}

\documentclass[border=2mm]{standalone}
\usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage[dvipsnames]{xcolor}
\usepackage{tikz-cd}

\begin{document}

\begin{tikzcd}[row sep=huge, column sep=huge,
        every label/.append style={font=\normalsize}]
    X_{11} \arrow[d, dashed] \arrow[rr,"{\textsf{ABC}}"]                          &  & X_{21}  \arrow[d, dashed]   \\
    X_{21} \arrow[d,"{\textsf{UVW}}", dashed] \arrow[rr, "{\textsf{DEF}}"]                          &  & X_{22}\arrow[d, dashed]     \\
    X_{31}  \arrow[rr, "{\textsf{XYZ}}", red] &  & X_{32}
\end{tikzcd}

\end{document}

\documentclass[border=2mm]{standalone}
\usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage[dvipsnames]{xcolor}
\usepackage{tikz-cd}

\begin{document}

\begin{tikzcd}[row sep=huge, column sep=huge,
        every label/.append style={font=\normalsize}]
    \mathcal{P}{}_h^\varepsilon \arrow[d,"{\scriptstyle\textsf{$h \to 0$}}" swap, dashed] \arrow[r, "{\scriptstyle\textsf{$\varepsilon \to 0$}}"]   &  \mathcal{P}{}_h^0 \arrow[d,"{\scriptstyle\textsf{$h \to 0$}}", dashed]
    \\
   \mathcal{P}{}^\varepsilon   \arrow[r, "{\scriptstyle\textsf{$\varepsilon \to 0$}}" swap] &  \mathcal{P}{}^0
\end{tikzcd}

\end{document}

流程图

\documentclass[border=2mm]{standalone}

\usepackage{tikz}
\usetikzlibrary{arrows.meta, positioning}

\begin{document}
\begin{tikzpicture}[
  node/.style={
    draw, rounded corners=5pt,
    minimum width=1.8cm, minimum height=0.9cm,
    font=\small\bfseries, align=center, thick,
    fill=blue!15
  },
  note/.style={
    font=\scriptsize\itshape, text=gray,
    align=center
  },
  edge/.style={-{Stealth[length=5pt]}, thick},
  annotate/.style={-{Stealth}, thick, teal!60!black, dashed},
  node distance=1.0cm and 1.3cm,
]

  % Main nodes: A → B → C → D
  \node[node, fill=orange!15] (A) {A};
  \node[node, right=of A] (B) {B};
  \node[node, right=of B] (C) {C};
  \node[node, right=of C, fill=teal!15] (D) {D};

  % Node notes (below each node)
  \node[note, below=2pt of A] {Input\\Stage};
  \node[note, below=2pt of B] {Processing\\Stage};
  \node[note, below=2pt of C] {Refinement\\Stage};
  \node[note, below=2pt of D] {Output\\Stage};

  % Main edges with labels
  \draw[edge, blue!60!black] (A) -- node[above, font=\footnotesize] {label 1} (B);
  \draw[edge, blue!60!black] (B) -- node[above, font=\footnotesize] {label 2} (C);
  \draw[edge, blue!60!black] (C) -- node[above, font=\footnotesize] {label 3} (D);

  % External annotation — dashed arrow from above pointing to node C
  \node[above=0.8cm of C, font=\scriptsize, teal!70!black] (ann) {External\\Factor};
  \draw[annotate] (ann) -- node[right, font=\tiny] {acts on} (C);

\end{tikzpicture}
\end{document}

\documentclass[border=2mm]{standalone}

\usepackage{tikz}
\usetikzlibrary{arrows.meta, positioning, decorations.pathreplacing}

\begin{document}
\begin{tikzpicture}[
  box/.style={
    draw, rounded corners=3pt,
    minimum width=1.8cm, minimum height=0.7cm,
    font=\small, align=center,
    fill=blue!8
  },
  arr/.style={-{Stealth}, thick},
  node distance=0.6cm,
]

  \node[box] (A) {Step 1};
  \node[box, below=of A] (B) {Step 2};
  \node[box, below=of B] (C) {Step 3};

  \draw[arr] (A) -- (B);
  \draw[arr] (B) -- (C);

  % Brace on left, mirror curves toward boxes
  \draw[decorate, decoration={brace, amplitude=8pt, raise=4pt, mirror}, thick, teal!60!black]
    (A.north west) -- (C.south west)
    node[midway, left=16pt, font=\small\bfseries, teal!60!black]
      {Pipeline};

\end{tikzpicture}
\end{document}

\documentclass[border=2mm]{standalone}

\usepackage{tikz}
\usetikzlibrary{arrows.meta, positioning}

\begin{document}
\begin{tikzpicture}[
  box/.style={
    draw, rounded corners=4pt,
    minimum width=2.0cm, minimum height=0.8cm,
    font=\small, align=center,
    fill=blue!10
  },
  arr/.style={-{Stealth}, thick, blue!60!black},
  node distance=0.7cm,
]
  \node[box, fill=orange!15] (A) {Input};
  \node[box, right=of A] (B) {Process\\A};
  \node[box, right=of B] (C) {Process\\B};
  \node[box, right=of C] (D) {Process\\C};
  \node[box, right=of D, fill=teal!15] (E) {Output};

  \draw[arr] (A) -- (B);
  \draw[arr] (B) -- (C);
  \draw[arr] (C) -- (D);
  \draw[arr] (D) -- (E);
\end{tikzpicture}
\end{document}

\documentclass[border=2mm]{standalone}

\usepackage{tikz}
\usetikzlibrary{arrows.meta, positioning}

\begin{document}
\begin{tikzpicture}[
  box/.style={
    draw, rounded corners=4pt,
    minimum width=2.0cm, minimum height=0.8cm,
    font=\small, align=center,
    fill=blue!10
  },
  arr/.style={-{Stealth}, thick, blue!60!black},
  node distance=0.7cm,
]
  \node[box, fill=orange!15] (A) {Input};
  \node[box, below=of A] (B) {Process\\A};
  \node[box, below=of B] (C) {Process\\B};
  \node[box, below=of C] (D) {Process\\C};
  \node[box, below=of D, fill=teal!15] (E) {Output};

  \draw[arr] (A) -- (B);
  \draw[arr] (B) -- (C);
  \draw[arr] (C) -- (D);
  \draw[arr] (D) -- (E);
\end{tikzpicture}
\end{document}

\documentclass[border=2mm]{standalone}

\usepackage{tikz}
\usetikzlibrary{arrows.meta, positioning}

\begin{document}
\begin{tikzpicture}[
  box/.style={
    draw, rounded corners=3pt,
    minimum width=3.2cm, minimum height=0.7cm,
    font=\small, align=center,
    fill=blue!8
  },
  excl/.style={
    draw, rounded corners=3pt,
    minimum width=2.4cm, minimum height=0.55cm,
    font=\scriptsize, align=center,
    fill=orange!10
  },
  arr/.style={-{Stealth}, thick},
]
  \node[box] (A) {Total Records\\$n = 1000$};
  \node[box, below=0.8cm of A] (B) {Screened\\$n = 920$};
  \node[box, below=0.8cm of B] (C) {Final Cohort\\$n = 880$};

  \node[excl, right=1.5cm of A] (E1) {Excluded:\\Criteria A\\$n = 80$};
  \node[excl, right=1.5cm of B] (E2) {Excluded:\\Criteria B $n=30$\\Criteria C $n=10$};

  \draw[arr] (A) -- (B);
  \draw[arr] (B) -- (C);
  \draw[arr] (A) -- (E1);
  \draw[arr] (B) -- (E2);
\end{tikzpicture}
\end{document}

\documentclass[border=2mm]{standalone}

\usepackage{tikz}
\usetikzlibrary{arrows.meta, positioning}

\begin{document}
\begin{tikzpicture}[
  box/.style={
    draw, rounded corners,
    minimum width=2.6cm, minimum height=0.7cm,
    font=\small, align=center,
    fill=blue!8
  },
  sub/.style={
    draw, rounded corners,
    minimum width=2.6cm, minimum height=0.7cm,
    font=\small, align=center,
    fill=teal!12
  },
  arr/.style={-{Stealth}, thick},
]

  \node[box] (main) {Main\\Process};

  \node[sub, below left=1.2cm and 0.5cm of main] (left) {Sub-process\\A};
  \node[sub, below right=1.2cm and 0.5cm of main] (right) {Sub-process\\B};

  \node[box, below=3.0cm of main] (result) {Final\\Result};

  \draw[arr] (main) -- (left);
  \draw[arr] (main) -- (right);
  \draw[arr] (left) -- (result);
  \draw[arr] (right) -- (result);

  \node[below=2pt of left, font=\scriptsize, text=gray] {method A};
  \node[below=2pt of right, font=\scriptsize, text=gray] {method B};

\end{tikzpicture}
\end{document}

\documentclass[border=2mm]{standalone}

\usepackage{tikz}
\usetikzlibrary{arrows.meta, positioning}

\begin{document}
\begin{tikzpicture}[
  box/.style={
    draw, rounded corners,
    minimum width=2.2cm, minimum height=0.8cm,
    font=\small, align=center,
    fill=blue!8
  },
  tool/.style={
    draw, rounded corners,
    minimum width=2.2cm, minimum height=0.8cm,
    font=\small, align=center,
    fill=teal!12
  },
  arr/.style={-{Stealth}, thick},
]

  \node[box] (core) {Core\\Engine};
  \node[tool, right=2.5cm of core] (parse) {Parse\\Command};
  \node[tool, below=1.2cm of parse] (exec) {Execute\\Action};
  \node[box, left=2.5cm of exec] (update) {Update\\State};

  \draw[arr] (core) -- node[above, font=\scriptsize] {output} (parse);
  \draw[arr] (parse) -- node[right, font=\scriptsize] {request} (exec);
  \draw[arr] (exec) -- node[below, font=\scriptsize] {response} (update);
  \draw[arr] (update) -- node[left, font=\scriptsize] {next iter} (core);

  \node[above=3pt of core, font=\scriptsize\itshape, text=blue!60!black] {internal};
  \node[below=3pt of exec, font=\scriptsize\itshape, text=teal!70!black] {external};

\end{tikzpicture}
\end{document}

\documentclass[border=2mm]{standalone}

\usepackage{tikz}
\usetikzlibrary{arrows.meta, positioning, shapes.geometric}

\begin{document}
\begin{tikzpicture}[
  box/.style={
    draw, rounded corners=3pt,
    minimum width=2.0cm, minimum height=0.7cm,
    font=\small, align=center,
    fill=blue!8
  },
  arr/.style={-{Stealth}, thick},
]

  \node[box] (start) {Start};

  \node[diamond, draw, aspect=2, inner sep=1pt,
        font=\small, fill=orange!10,
        below=0.9cm of start] (check) {Check\\OK?};

  \node[box, right=2.0cm of check] (retry) {Adjust};
  \node[box, below=1.2cm of check] (done) {Done};

  \draw[arr] (start) -- (check);
  \draw[arr] (check) -- node[above, font=\scriptsize] {No} (retry);
  \draw[arr] (retry.east) -- ++(0.6,0) |- (start.east);
  \draw[arr] (check) -- node[right, font=\scriptsize] {Yes} (done);

\end{tikzpicture}
\end{document}

\documentclass[border=2mm]{standalone}

\usepackage{tikz}
\usetikzlibrary{arrows.meta}

\begin{document}
\begin{tikzpicture}[
  milestone/.style={thick},
  label above/.style={above=8pt, font=\small\bfseries},
  label below/.style={below=8pt, font=\footnotesize\itshape, text=gray},
]

  % Main axis
  \draw[-{Stealth}, thick, blue!60!black] (0,0) -- (12,0) node[right] {};

  % Events
  \foreach \x/\name/\desc in {
    1.5/Phase A/Preparation,
    5.0/Phase B/Execution,
    9.0/Phase C/Delivery%
  } {
    \draw[milestone, blue!60!black] (\x, 0.15) -- (\x, -0.15);
    \fill[blue!60!black] (\x, 0) circle (3pt);
    \node[label above, blue!60!black] at (\x, 0.15) {\name};
    \node[label below] at (\x, -0.15) {\desc};
  }

\end{tikzpicture}
\end{document}

数据结构

\documentclass[border=2mm]{standalone}
\usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage[dvipsnames]{xcolor}
\usepackage{tikz}

\begin{document}

\begin{tikzpicture}
    % Nodes
    \node[circle, draw] (A) at (0, 0) {A};
    \node[circle, draw] (B) at (2, 2) {B};
    \node[circle, draw] (C) at (2.3, 0) {C};
    \node[circle, draw] (D) at (2, -2) {D};
    \node[circle, draw] (F) at (4, 1) {F};
    \node[circle, draw] (E) at (4, -1) {E};

    % Edges with weights
    \draw (A) -- (B) node[midway, above] {9};
    \draw (A) -- (C) node[midway, above] {9};
    \draw (A) -- (D) node[midway, below left] {7};
    \draw (B) -- (C) node[midway, right] {2};
    \draw (C) -- (D) node[midway, left] {1};
    \draw (B) -- (F) node[midway, above] {9};
    \draw (C) -- (E) node[midway, below left] {11};
    \draw (D) -- (E) node[midway, below right] {15};
    \draw (E) -- (F) node[midway, right] {6};

    % Highlight the shortest path
    \draw[orange, thick] (A) -- (D);
    \draw[orange, thick] (D) -- (C);
    \draw[orange, thick] (C) -- (B);
    \draw[orange, thick] (B) -- (F);
\end{tikzpicture}

\end{document}

\documentclass[border=2mm]{standalone}
\usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage[dvipsnames]{xcolor}
\usepackage{tikz}

\begin{document}

\begin{tikzpicture}[scale=1.2]
    \node[] (root) at (0,0) {$\{x_1,v_1,x_2,v_2,x_3,v_3\}$};
    \node (x1v1x2v2) at (-2,-1) {$\{x_1,v_1,x_2,v_2\}$};
    \node (x3v3) at (1.5,-1) {$\{x_3,v_3\}$};

    \node (x1v1) at (-3,-2) {$\{x_1,v_1\}$};
    \node (x2v2) at (-1,-2) {$\{x_2,v_2\}$};

    \node[red] (x1) at (-3.5,-3) {$\{x_1\}$};
    \node[red] (v1) at (-2.5,-3) {$\{v_1\}$};
    \node[red] (x2) at (-1.5,-3) {$\{x_2\}$};
    \node[red] (v2) at (-0.5,-3) {$\{v_2\}$};
    \node[red] (x3) at (0.75,-2) {$\{x_3\}$};
    \node[red] (v3) at (2.25,-2) {$\{v_3\}$};

    \draw[->] (root) -- (x1v1x2v2);
    \draw[->] (root) -- (x3v3);
    \draw[->] (x1v1x2v2) -- (x1v1);
    \draw[->] (x1v1x2v2) -- (x2v2);
    \draw[->] (x1v1) -- (x1);
    \draw[->] (x1v1) -- (v1);
    \draw[->] (x2v2) -- (x2);
    \draw[->] (x2v2) -- (v2);
    \draw[->] (x3v3) -- (x3);
    \draw[->] (x3v3) -- (v3);
\end{tikzpicture}

\end{document}

\documentclass[border=2mm]{standalone}

\usepackage{tikz}
\usetikzlibrary{arrows.meta}

\begin{document}
\begin{tikzpicture}[
  level 1/.style={level distance=1.4cm, sibling distance=4.0cm},
  level 2/.style={level distance=1.4cm, sibling distance=1.8cm},
  every node/.style={
    draw, rounded corners=3pt,
    font=\small, minimum width=1.5cm, align=center,
    fill=blue!8
  },
  edge from parent/.style={draw, -{Stealth}, thick},
]

  \node {Root}
    child {
      node {Branch A}
      child { node {Leaf A1} }
      child { node {Leaf A2} }
    }
    child {
      node {Branch B}
      child { node {Leaf B1} }
      child { node {Leaf B2} }
    };

\end{tikzpicture}
\end{document}

关于树结构，还可以使用专门的宏包 forest

\documentclass[border=2mm]{standalone}
\usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage[dvipsnames]{xcolor}
\usepackage{tikz}

\begin{document}

\begin{tikzpicture}
    \draw[] (-4, -2.5) rectangle (4, 2.5) node[below left] {U};
    \draw[] (-1,-.3) circle (2) node[right=2cm] {$V$};
    \node[] at (-1.3,0.8) {a};
    \node[] at (0.1,0.6) {e};
    \node[] at (-1.6,-0.5) {i};
    \node[] at (0,-1) {o};
    \node[] at (-1,-1.6) {u};
\end{tikzpicture}

\medskip
\begin{tikzpicture}
    \draw[] (-4, -2.5) rectangle (4, 2.5) node[below left] {U};
    \path[opacity=.3, even odd rule]
    (-1,-.3) circle (2)
    (1,-.3) circle (2);

    \begin{scope}
        \clip (-1,-.3) circle (2);
        \fill[opacity=.3] (1,-.3) circle (2);
    \end{scope}

    \draw[] (-1,-.3) circle (2) node[above=2cm] {$A$};
    \draw[] (1,-.3) circle (2) node[above=2cm] {$B$};
\end{tikzpicture}

\medskip
\begin{tikzpicture}
    \draw[] (-4, -2.5) rectangle (4, 2.5) node[below left] {U};

    \begin{scope}
        \clip (-1,-.3) circle (2);
        \fill[opacity=.3] (1,-.3) circle (2);
    \end{scope}

    \draw[] (-1,-.3) circle (2) node[above=2cm] {$A$};
    \draw[] (1,-.3) circle (2) node[above=2cm] {$B$};
\end{tikzpicture}

\end{document}

% This file is from Weihao Xia, Jing-Hao Xue. A Survey on 3D-aware Image Synthesis. https://arxiv.org/abs/2210.14267

\documentclass[letterpaper]{article}
\usepackage{tikz}
\usepackage{longtable}
\usepackage{tikz-qtree}
\usepackage[edges]{forest}
\usetikzlibrary{trees}


\begin{document}


\begin{figure}[t!]
    \centering
    \tikzset{
        my node/.style={
                font=\small,
                rectangle,
                draw=#1!75,
                align=justify,
            }
    }
    \forestset{
        my tree style/.style={
                for tree={grow=east,
                        parent anchor=east, %
                        child anchor=west,
                        where level=0{my node=black,text width=1.5em}{},
                        where level=1{my node=black,text width=8.5em}{},
                        where level=2{my node=black,text width=8.0em}{},
                        where level=3{my node=black,text width=15em}{},
                        l sep=1.5em,
                        forked edge,                %
                        fork sep=1em,               %
                        edge={draw=black!50, thick},
                        if n children=3{for children={
                                        if n=2{calign with current}{}}
                            }{},
                        tier/.option=level,
                    }
            }
    }
    \begin{forest}
        my tree style
        [All
        [3D Generative Models from Single Views
            [Conditional 3D GANs
                    [Semantic Label]
                    [Image]
            ]
            [Unconditional 3D GANs
                    [Interactive Editing]
                    [Efficient and Consistent Rendering
                    ]
                    [Efficient and Effective Representations]
            ]
        ]
        [3D Novel View Synthesis from Multiple Views
        [Constrained $\rightarrow$ In-the-wild
        [Static $\rightarrow$ Dynamic]
        [Single $\rightarrow$ Large-scale]
        [Calibrated $\rightarrow$ Uncalibrated]
        [Dense $\rightarrow$ Sparse]
        ]
        [Speed up]
        [Neural Scene Representation]
        ]
        [3D Control of 2D Generative Models
            [3D Priors as Constraints
                    [3D Components into 2D GANs]
                    [3D Prior Knowledge]
            ]
            [3D Parameters as Controls
                    [Explicit Control over 3D Parameters]
                    [Control Factors from Pretrained Models]
            ]
            [3D Control Latent Directions
                    [Unsupervised]
                    [Supervised]
            ]
        ]
        ]
    \end{forest}
    \centering
    \caption{A systematic taxonomy.}
    \label{fig:phylogenetic_tree}
\end{figure}

\end{document}

补充：PDF 转 PNG

提供一个基于 gs 将 PDF 格式的图片转换为 PNG 的脚本。

#!/bin/bash
# 将当前目录下所有 PDF 转成高分辨率 PNG，每页单独输出，保留原文件名

DPI=600

# 遍历当前目录下所有 PDF 文件
for pdf in *.pdf; do
    # 去掉扩展名
    filename="${pdf%.pdf}"
    # 使用 Ghostscript 转 PNG，每页生成一个文件，带页码
    gs -dSAFER -dBATCH -dNOPAUSE -sDEVICE=png16m -r$DPI -sOutputFile="${filename}-%d.png" "$pdf"
    echo "Converted $pdf -> ${filename}-%d.png"
done

echo "All PDFs converted to high-resolution PNG."