以下的分析来自 codex-5.5-xhigh:

原因是 book.cls 的 \tableofcontents 把 \@mkboth 放进了 \chapter*{...} 标题参数里；你的 \titleformat 又把该标题交给 \myroundedbox'，其中 xstring' 的 \StrLeft/\StrGobbleLeft 会拆解整个 token list，触发 \@mkboth 递归，最终报 TeX capacity exceeded [input stack size=10000]。
我已采用最佳修正：重定义 \tableofcontents，把 \@mkboth 移到 \chapter*{\contentsname} 之后执行。这样目录标题装饰宏只处理可见文字“目录”，不再处理页眉标记命令。

\documentclass[fontset=fandol]{ctexbook}
\RequirePackage{titlesec}
\RequirePackage{fancyhdr}
\RequirePackage{xstring}
\RequirePackage{tikz}
\newcommand{\myroundedbox}[1]{%
  \begin{tikzpicture}[baseline=(mbox.base), overlay, remember picture]
    \StrLeft{#1}{1}[\firstchar]
    \StrGobbleLeft{#1}{1}[\restchars]
    \node[anchor=west,draw=cyan,line width=2pt,fill=white,inner xsep=3pt,inner ysep=3pt,font=\bfseries] (mbox) {\firstchar\hskip 0.3em\restchars};
  \end{tikzpicture}%
}
\titleformat{\chapter}[display]
  {%
    \sffamily%
    \thispagestyle{empty}
  }
  {}
  {-1.1em}
  {\myroundedbox}
\titlespacing{\chapter}{0pt}{-1.5\baselineskip}{0\baselineskip}
% book.cls puts \@mkboth inside the TOC chapter title; keep it outside
% because \myroundedbox uses xstring to split only visible title text.
\makeatletter
\renewcommand\tableofcontents{%
    \if@twocolumn
      \@restonecoltrue\onecolumn
    \else
      \@restonecolfalse
    \fi
    \chapter*{\contentsname}%
    \@mkboth{\MakeUppercase\contentsname}{\MakeUppercase\contentsname}%
    \@starttoc{toc}%
    \if@restonecol\twocolumn\fi
    }
\makeatother

\begin{document}

\tableofcontents %

\chapter{课节导引}
\section{引导点公式速记速记}

\end{document}

如上的 patch 可以似乎可以解决问题...

Claim:这不是一个回答，只是必要的简化。

该问题的一个简化版本等价于：

\documentclass[fontset=fandol]{ctexbook}
\RequirePackage{titlesec}
\RequirePackage{fancyhdr}
\RequirePackage{xstring}
\RequirePackage{tikz}
\newcommand{\myroundedbox}[1]{%
  \begin{tikzpicture}[baseline=(mbox.base), overlay, remember picture]
    \StrLeft{#1}{1}[\firstchar]
    \StrGobbleLeft{#1}{1}[\restchars]
    \node[anchor=west,draw=cyan,line width=2pt,fill=white,inner xsep=3pt,inner ysep=3pt,font=\bfseries] (mbox) {\firstchar\hskip 0.3em\restchars};
  \end{tikzpicture}%
}
\titleformat{\chapter}[display]
  {%
    \sffamily%
    \thispagestyle{empty}
  }
  {}
  {-1.1em}
  {\myroundedbox}
\titlespacing{\chapter}{0pt}{-1.5\baselineskip}{0\baselineskip}

\begin{document}

% \tableofcontents %

\chapter{课节导引}
\section{引导点公式速记速记}

\end{document}

如果加上\tableofcontents，将导致如下报错：

! TeX capacity exceeded, sorry [input stack size=10000].
\@mkboth ...rotect \ttl@gmk {\protect \@mkboth {#1
                                                  }{#2}}
l.25 \tableofcontents

另外，对于这种站内的链接，不建议不贴文件。否则这隐含着让其他人都要付费下载一次。

能否请 @u122987 关注一下。

Code1. 源代码

\documentclass[zihao=-4]{ctexbook}
\RequirePackage[
    papersize={210mm,297mm},
    margin=25mm,showframe,
    ]{geometry} 
\RequirePackage{enumitem} 

\usepackage{newtxtext}

\begin{document}

\begin{enumerate}[label={[}\arabic*{]}, labelindent=\parindent, leftmargin=*]
    \item Tao Y, \textbf{Liu H}, Wang X, et al. Probabilistically Informed Motion Primitives for Impedance Control with Guided Motion Planning in Contact-Rich Manipulation Tasks[J]. xxxx. (学生一作在投，SCI，Q1, IF=4.7，对应第四章)
\end{enumerate}

\end{document}

A. xelatex+newtxtext

B. lualatex+newtxtext

Code2.添加\usepackage[british]{babel}

\documentclass[zihao=-4]{ctexbook}
\RequirePackage[
    papersize={210mm,297mm},
    margin=25mm,showframe,
    ]{geometry} 
\RequirePackage{enumitem} 
\usepackage[british]{babel}
\usepackage{newtxtext}

\begin{document}

\begin{enumerate}[label={[}\arabic*{]}, labelindent=\parindent, leftmargin=*]
    \item Tao Y, \textbf{Liu H}, Wang X, et al. Probabilistically Informed Motion Primitives for Impedance Control with Guided Motion Planning in Contact-Rich Manipulation Tasks[J]. xxxx. (学生一作在投，SCI，Q1, IF=4.7，对应第四章)
\end{enumerate}

\end{document}

A. xelatex+newtxtext+\usepackage[british]{babel}

B. lualatex+newtxtext+\usepackage[british]{babel}

Code3.使用\hyphenation{im-ped-ance}

\documentclass[zihao=-4]{ctexbook}
\RequirePackage[
    papersize={210mm,297mm},
    margin=25mm,showframe,
    ]{geometry} 
\RequirePackage{enumitem} 
\hyphenation{im-ped-ance}
\usepackage{newtxtext}

\begin{document}

\begin{enumerate}[label={[}\arabic*{]}, labelindent=\parindent, leftmargin=*]
    \item Tao Y, \textbf{Liu H}, Wang X, et al. Probabilistically Informed Motion Primitives for Impedance Control with Guided Motion Planning in Contact-Rich Manipulation Tasks[J]. xxxx. (学生一作在投，SCI，Q1, IF=4.7，对应第四章)
\end{enumerate}

\end{document}

A.xelatex+\hyphenation{im-ped-ance}:

B.lualatex+\hyphenation{im-ped-ance}:

Code4.添加microtype(但只在lualatex下生效，感谢@u30 )

% lualatex
\documentclass[zihao=-4]{ctexbook}
\RequirePackage[
    papersize={210mm,297mm},
    margin=25mm,showframe,
    ]{geometry} 
\RequirePackage{enumitem} 
\usepackage{newtxtext}
\usepackage{microtype} 

\begin{document}

\begin{enumerate}[label={[}\arabic*{]}, labelindent=\parindent, leftmargin=*]
    \item Tao Y, \textbf{Liu H}, Wang X, et al. Probabilistically Informed Motion Primitives for Impedance Control with Guided Motion Planning in Contact-Rich Manipulation Tasks[J]. xxxx. (学生一作在投，SCI，Q1, IF=4.7，对应第四章)
\end{enumerate}

\end{document}

https://tex.stackexchange.com/a/762259/322482

我试了一下，虽然没有任何提示，但是在这个网页 只要耐心等一分钟就可以触发自动下载。

但为何在macOS下，却无法在终端中使用makeglossaries命令编译词汇表，提示找不到命令

终端能找到系统程序的必要条件是 makeglossaries.exe 需要在系统的 $PATH 路径下，可以尝试一下环境变量路径下是否真的有 makeglossaries.exe 这个程序。

\documentclass[fontset=fandol]{ctexbeamer}
% Refs: https://tex.stackexchange.com/a/380895
% \usepackage{multicol}
\xdefinecolor{mycolor}{RGB}{0,102,204}
\begin{document}
{%
\setbeamertemplate{headline}{}
\setbeamertemplate{footline}{}
\begin{frame}[plain]%
    % \vspace*{\fill}
    \centering
    {\Huge \bfseries \textcolor{mycolor}{目录}}%
    \vspace{1.2em}%
    \textcolor{mycolor}{\hrule}
    \vspace{1.5em}%
    % \begin{minipage}{0.85\textwidth}%
    %     \begin{multicols}{2}
    %         \tableofcontents[hideallsubsections]
    %     \end{multicols}
    % \end{minipage}
    \begin{columns}
        \begin{column}{.3\textwidth}%
            \tableofcontents[sections=1-3,hideallsubsections]
        \end{column}%
        \begin{column}{.3\textwidth}%
            \tableofcontents[sections=4-6,hideallsubsections]
        \end{column}%
    \end{columns}%
\end{frame}%
}
\section{第一章 研究背景}
\section{第二章 研究意义}
\section{第三章 研究内容}
\section{第四章 研究方法}
\section{第五章 技术路线}
\section{第六章 总结展望}

\begin{frame}
    正文
\end{frame}
\end{document}

BTW, xdefinecolor好像很少见...

请用「CTeX套装」试试而不是「TeXLive/Miktex」。同时如果还不行，请详细说明你具体是怎么做的，报错信息是什么？

Refs: https://ask.latexstudio.net/ask/question/17778.html

很简单的办法参考自： https://ask.latexstudio.net/ask/question/8198.html

对于这种分段函数，直接使用tikz用直线绘制即可，你甚至可以很方便的控制颜色，线条样式、粗细等特征：

\documentclass[tikz,border=5pt]{standalone}
\usepackage[fontset=fandol]{ctex}
\usepackage[osf]{libertine} % 我喜欢的一款字体
\begin{document}

\begin{tikzpicture}[
    line cap=round,
    every node/.style={black}
    ]
\draw[-latex] (0,0) -- (6,0) node[below]{t(h)};
\draw[-latex] (0,0) -- (0,6) node[right]{s(km)};

\node[below left] {0};

\foreach \i in {1,...,5}{
    \draw(0,\i) node[left] {\inteval{10*\i}} -- ++(.2,0);
    \draw(\i,0) node[below] {\inteval{\i}} -- ++(0,.2);
}

\draw[magenta,thick] (0,0) 
    -- (1,3) node[left] {A} 
    -- (1.5,3) node[right] {B} 
    -- (2,5) node[above left=-1pt] {C} 
    -- (2.75,5) node[above right=-1pt] {D} 
    -- (5,0) node[above right=-1pt] {E}
    ;
\draw[cyan,dashed,semithick] (0,5) 
    -- (.5,5) node[above] {G}
    -- (3,0) node[above right] {H}
    ;

\draw[cyan,dashed,semithick] (5.75,5.25) -- ++(-1,0) node[left,anchor=east] {小聪};
\draw[magenta,thick] (5.75,4.5) -- ++(-1,0) node[left,anchor=east] {小慧};

\end{tikzpicture}

\end{document}

当然，这里有个小难点是如何恰好让「B」点相交，我是做了一些斜率的预计算。放轻松，每一行命令的功能都是比较简单的，你可以注释某行代码再编编译看看结果，相信学会这种绘制并不算太难~

Happy TikZing!

使用chemfig宏包酚醛树脂的缩聚反应方程式如何写，达到图片上的效果，特别是方括号及高分子连线那部分

具体是哪里不会呢？看上去是只有方括号和连线没处理好，请展示你现在已有的代码，避免问「do-this-for-me类」的问题。

不是非常优雅的一些暴力微调...

\documentclass{article}%
\usepackage{newpxmath}%
\usepackage{chemfig}
\usepackage{mhchem}
\renewcommand*\printatom[1]{\ensuremath{\mathsf{#1}}}
\begin{document}
\[
    \setchemfig{atom sep=2em}
    \hreac%[hreac debug]
        \^{2pt}$n$\;\chemfig[baseline=10pt]{*6([,0.75]=-=(-[0]CH_2OH)-(-[2]OH)=-)}
        >[{\ce{H+}}][{$\triangle$}]
        \^{2pt}\chemfig{H-[@{op,.5}]([:-30]*6([,0.75]-=-=(-[0]CH_2-[0]@{cl,0}-[0]OH)-(-[2]OH)=))}
        \polymerdelim[delimiters ={[]}, height = 5pt, depth=22.5pt]{op}{cl}
        + $(n - 1)$\ce{H2O}
        \endhreac
\]
\end{document}

Just refer the manual in texdoc nicematrix and a little bit in texdoc tikz:

\documentclass{article}
\usepackage{fourier}
\usepackage{nicematrix,tikz}
\usepackage{lipsum}
\usetikzlibrary{arrows.meta}
\begin{document}

\lipsum[2][1-3]

\[  
\renewcommand{\arraystretch}{1.5}%
\begin{pNiceArray}{ccc;ccc}[margin] % new feature `;' in nicematrix v7.7
\CodeBefore%
\tikz{%
    \draw[-Stealth] ([xshift=-1cm,yshift=.05cm]1.5|-2) node[above] {$(-2)$} |- ([xshift=-.5cm]1.5|-2.5);
    \draw[-Stealth] ([xshift=-1.75cm,yshift=.05cm]1.5|-2) node[above] {$(-1)$} |- ([xshift=-.5cm]1.5|-3.5);
}
\Body
1 & 2 & 3 & 1 & 0 & 0 \\
2 & 1 & 2 & 0 & 1 & 0 \\
3 & 2 & 1 & 0 & 0 & 1 \\
\end{pNiceArray}
\]

\lipsum[2][1-3]

\end{document}

@u444 代码要尽量给完整，不要只给一个片段，同时要放在代码块内。

注意到原图其实是想表达3D的角度，原图的展示不符合实际情况，你在评论区中给出的图又完全是2D的，也似乎不完全符合实际情况。

以下是基于luadraw的一个可能方案，可以解决「但是代码不够好看以及视角不够灵活等缺点」，需要自行安装并用lualatex编译：

\documentclass{standalone}
\usepackage[3d]{luadraw}
\usepackage[svgnames]{xcolor}
\usepackage{fourier}
\begin{document}
\begin{luadraw}{name=axis_vector}
local g = graph3d:new{ 
    window3d={-1,9,-1,9,-1,9}, 
    window = {-6,8,-5,8},size={12,12},
    viewdir=perspective("xy",0.8,60), 
    pictureoptions="every node/.append style={scale=1.5}"
}
local O,delta,P = Origin,-3, M(6.5, 6, 4)
local Pxz = pxz(P)
local Px,Pz,Py = px(Pxz), pz(Pxz), py(P)
local t1, t2, t3 = M(delta, 0, P.z), M(delta,0, 0), M(delta, P.y, 0)

g:Dpolyline3d({{O, 7*vecI}, {O, 7*vecJ}, {O, 7*vecK}}, "-Stealth,solid,black, line width=1pt")
g:Dlabel3d(
    "$x$", 7*vecI, {pos="N",dist=0.1}, 
    "$y$", 7*vecJ, {pos="W"}, 
    "$z$", 7*vecK, {pos="E"}
    )

g:Dpolyline3d({{P,Py}, {P,Pxz}, {Pxz,Px}, {Pxz,t1},{O,Pxz}}, "dashed,blue,thick")
g:Dpolyline3d({{O,t2}, {Py,t3}}, "dashed,black,thick")

g:Dpolyline3d({{t1,t2}, {t2,t3}, {Pxz,Pz}}, "Stealth-Stealth,dashed,magenta,thick")
g:Dlabel3d(
    "$z$", (t1+t2)/2, {pos="NW",node_options="text=magenta"},
    "$y$", (t2+t3)/2, {pos="W"},
    "$x$", (Pxz+Pz)/2, {pos="S"}
)

g:Dseg3d({O, P}, "-latex,PaleVioletRed,line width=4pt")
g:Dpolyline3d({{O, 2.5*vecI}, {O, 2.5*vecJ}, {O, 2.5*vecK}}, "-latex,cyan,line width=3pt")
g:Dlabel3d("$\\vec{i}$", 2.5*vecI, {pos="S",node_options="text=cyan"}, "$\\vec{j}$", 2.5*vecJ, {pos="W"}, "$\\vec{k}$", 2.5*vecK, {})

g:Ddots3d({P}, "PaleVioletRed,scale=4", 1.5)
g:Dlabel3d("$P$", P, {pos="NE"},"$\\vec{r}$", P, {pos="W",dist=0.35,node_options="text=PaleVioletRed"})

for _,i in ipairs({Px, Py, Pz}) do
    g:Ddots3d({i}, "blue", 1.5)
end

local configs = {
    {pt=Px, col="blue"},
    {pt=Py, col="green"},
    {pt=Pz, col="violet"}
}
for _, item in ipairs(configs) do
    g:Dpath3d({item.pt, O, P, 3.75, 1, "ca", O, "l"}, "draw=none,opacity=0.2,fill=" .. item.col)
    g:Darc3d(item.pt, O, P, 3.75, 1, "-Stealth,thick," .. item.col .. ",ultra thick")
end

g:Dlabel(
    "$\\alpha$",Z(3.25,1), {node_options="text=blue"},
    "$\\beta$", Z(1.5,3), {node_options="text=green"},
    "$\\gamma$", Z(0.5,-1.2), {node_options="text=violet"}
)

g:Show()

\end{luadraw}
\end{document}

Here below is the proposal of luadraw:

Some details need to be tweaked by yrself.

% https://ask.latexstudio.net/ask/question/8020.html
\documentclass{standalone}
\usepackage[fontset=fandol]{ctex}
\usepackage[3d]{luadraw}
\usepackage[svgnames]{xcolor}
\usepackage{fourier}
\begin{document}

    \begin{luadraw}{name=earth_tilted_system}
        local g = graph3d:new{
            window3d={-8,8,-8,8,-6,6}, 
            window = {-8,9,-6,6},
            size={10,10}, viewdir={0,80},
            pictureoptions = "line cap=round"
        }
        -- first half plane
        g:Dpath3d(
            {M(0,-7,0),M(-8,-6,0), M(-8,8,0), M(0,7,0),"l", 4*vecJ, "m", Origin, -4*vecJ, 4, 1, vecK, "ca"},
            "fill=yellow!15, fill opacity=1, draw=orange, line width=0.5pt"
        )
        g:Darc3d(4*vecJ, Origin, -4*vecJ, 4, 1, vecK, "teal, line width=0.5pt") -- inner edge
        -- Rotate3d alter the transfoemmation matrix
        g:Saveattr()
            g:Rotate3d(-23.26, {Origin, vecI})
            g:Dcircle3d(Origin, 4, vecK, "fill=yellow, draw=orange, line width=0.5pt")
            g:Dsphere(Origin, 4, {mode=mBorder, edgecolor="cyan", edgewidth=10})
        g:Restoreattr()
        -- second half plane
        g:Dpath3d(
            {M(0,-7,0),M(8,-8,0), M(8,6,0), M(0,7,0),"l", 4*vecJ, "m", Origin, -4*vecJ, 4, -1, vecK, "ca"},
            "fill=yellow!15, fill opacity=1, draw=orange, line width=0.5pt"
        )
        g:Darc3d(4*vecJ, Origin, -4*vecJ, 4, -1, vecK, "teal, line width=0.5pt") -- outer edge
        -- hidden lines
        g:Dcircle3d(Origin, 4, g.Normal, "dashed,cyan,line width=0.8pt")
        g:Dcircle3d(Origin, 4, vecK, "dashed,teal")
        -- Rotate3d alter the transfoemmation matrix again
        g:Saveattr()
        g:Rotate3d(-23.26, {Origin, vecI})
        -- hidden ellipse
        -- g:Dcircle3d(Origin, 4, vecK, "orange,dashed")
        -- the z-axis
        g:Dline3d({-1.5 * vecK, 1.5 * vecK}, "black, dashed, line width=0.8pt")
        -- the rotated y-axis
        g:Dseg3d({Origin, 8 * vecJ}, "teal, dashed, line width=0.8pt")
        g:Dlabel3d(
            "赤道平面", M(0.5,-2,0), {dir={vecJ,M(-8,1,0)},node_options="scale=0.75"}
        )
        g:Restoreattr()
        -- magenta segments 
        g:Dseg3d({Origin, 4 * vecJ}, "magenta, dashed, line width=0.8pt")
        g:Dseg3d({8*vecJ, 4 * vecJ}, "magenta, line width=0.8pt")
        -- arc3d
        g:Darc3d(5 * vecJ, Origin, Ms(4,90*deg,23.26*deg),5,1, "red,latex-latex") -- 90*deg = pi/2 (radians)
        g:Darc3d(8*vecJ, Origin, Ms(4,90*deg,(90+23.26)*deg),8,1, "red,latex-latex")
        g:Dlabel3d(
            "黄道平面", M(5,-6,0), {dir={vecJ,M(-8,1,0)},node_options="scale=0.75"},
            "$66^\\circ 34'$", Ms(5,90*deg,(45+23.26/2)*deg), {
                node_options="fill=white,inner sep=1.5pt", 
                dir={Ms(1,90*deg,(135+23.26/2)*deg), Ms(1,90*deg,(45+23.26/2)*deg)}
            },
            "$23^\\circ 26'$", Ms(8,90*deg,(90+23.26/2)*deg), 
            {node_options="fill=white,inner sep=1.5pt", dir={Ms(1,90*deg,(180+23.26/2)*deg), Ms(1,90*deg,(90+23.26/2)*deg)}}
        )
        g:Dlabel(
            "黄\\\\赤\\\\交\\\\角", Z(8.75,-1.75,0), {node_options="scale=0.8,text=red,align=left,font=\\bfseries\\linespread{1}\\selectfont"}
        )
        g:Show()
    \end{luadraw}
\end{document}

hspan=minimal is all I need, actually...

对文档还是不够熟悉(摇头...)

\documentclass[12pt]{article}
\usepackage[fontset=fandol]{ctex}
\usepackage[showframe,a4paper,hmargin=1.25cm]{geometry}
\usepackage{tabularray}
\UseTblrLibrary{booktabs,varwidth}
\setlength{\parindent}{0pt}
\usepackage{lipsum}
\begin{document}

\begin{tblr}{
    colspec={X[1.25,c,m]X[1.75,c,m]X[1,c,m]X[1.5,c,m]X[1.75,c,m]},
    hlines,vlines,
    hline{1,Z} = {2pt},
    vline{1,Z} = {2pt},
    width=\linewidth,
    measure=vbox,
    hspan=minimal,
    rows = {ht=1cm},
    row{3} = {ht=2.5cm},
    row{X-Y} = {ht=3cm},
    cell{1}{1-Y} = {r=2}{valign=m},
    cell{2}{Z} = {r=3}{valign=m},
    cell{3}{2} = {c=3}{valign=m},
    cell{5-Z}{2} = {c=4}{halign=l,valign=m},
    cell{X-Y}{2} = {valign=h},
}
课程名称 & & {学\quad 分} & & {总计:\qquad 学时} \\
& & & & {\linespread{1.75}\selectfont 讲课:\qquad 学时\\讲课:\qquad 学时\\讲课:\qquad 学时\\讲课:\qquad 学时\\讲课:\qquad 学时\\} \\
{课程性质\\[10pt](请打勾)} & %
{\linespread{1.5}\selectfont%<- important % hete
    通识必修课（\quad）\ 通识选修课（\quad）\ 专业必修课（\quad）\\
    通识必修课（\quad）\ 通识选修课（\quad）\ 专业必修课（\quad）\\
    通识必修课（\quad）\ 通识选修课（\quad）\ 专业必修课（\quad）\\
}  
&  &  &  \\
任课老师 &  & 职称 &  &  \\
授课对象 & 年级、专业： &  &  &  \\
课程目标 &  &  &  &  \\
教学重难点 &  &  &  &  \\
% 教材 & {教材：} &  &  &  \\
% 参考资料 & {参考资料：} &  &  &  \\
教材 & {教材：\lipsum[2][1-6]} &  &  &  \\
参考资料 & {参考资料：\lipsum[2][1-6]} &  &  &  \\
其他要求 &  &  &  &  \\
\end{tblr}

\bigskip

Notes: \lipsum[2][1-5]

\end{document}

使用需要lualatex的tkz-elements包(v5.13c)在处理椭圆切线等问题会方便很多：

\documentclass[border=5pt]{standalone}
\usepackage[mini]{tkz-euclide}
\usepackage{tkz-elements}
\directlua{
    init_elements()
    z.O = point(0.15, 0.6) 
    C.OW = circle(through(z.O, 0.5))
    z.W = C.OW.through
    z.M = point(-0.25, -1)
    z.N = point(1,4)
    z.C = tkz.midpoint(z.M, z.N)
    L.MN = line(z.M, z.N)
    z.T = L.MN:orthogonal_at(z.C, {length = 1.5})
    CO.EL = conic(EL_points(z.C, z.N, z.T))
    PA.curve = CO.EL:points(0, 1, 100)
    z.P = point(4,-1)
    L.tan, _ = CO.EL:tangent_from(z.P)
    z.Q = L.tan.pb
    L.normal = L.tan:ortho_from(z.Q)
    z.dir = L.normal:report(-1.5, z.Q)
    z.H = line(z.P, z.Q):projection(z.O)
    z.v, _ = intersection(line(z.O,z.Q), C.OW)
    z.u, _ = intersection(line(z.O,z.H), C.OW)
}
\begin{document}

\begin{tikzpicture}
    \tkzGetNodes
    \tkzDrawCoordinates[smooth,thick](PA.curve)  \tkzDrawCircle[black,thick](O,W)
    \tkzDrawSegments[thick,-latex](Q,dir)
    \tkzDrawSegments[thick,add=0 and 0.5](P,Q)
    \tkzDrawSegments[thick,dashed,green](O,Q)
    \tkzDrawSegments[thick,green,-latex](O,v)
    \tkzDrawSegments[thick,dash dot,blue](O,H)
    \tkzDrawSegments[thick,blue,-latex](O,u)
    \tkzMarkRightAngle[size=.25,thick](O,H,Q)
    \tkzMarkRightAngle[size=.25,thick](P,Q,dir)
    \draw [magenta,decorate,decoration={brace,mirror,amplitude=10pt}] (O) -- (H) node[midway,below right,xshift=-.15cm,yshift=-.25cm,black] {$h(K,u)$};
    % labels left is your time~
\end{tikzpicture}

\end{document}