Chapter 11: test2

https://www.overleaf.com/learn/latex/Commands

\newcommand

\renewcommand to replace existing command

https://www.physicsread.com/latex-newcommand/

font color^[10.6.1]

$$\mathrm{sin^{}}\left(\alpha\right), \mathrm{sin^{n}}\left(\beta\right),\mathrm{sin^{m}}\left(\gamma\right)$$

$$\mathrm{cos}\left(\color{red}{2\theta}\right)-\mathrm{sin}\left(\color{blue}{2\theta}\right)=\mathrm{cos}\left(\color{green}{4\theta}\right)$$

$$\theta = \mathrm{tan^{-1}}\left(\color{red}{\frac{x}{y}}\right)$$

$$\mathrm{tan}\left(\color{red}{\alpha+\beta}\right) = \frac{\mathrm{tan}\left(\color{blue}{\alpha}\right) + \mathrm{tan}\left(\color{blue}{\beta}\right)}{1 - \mathrm{tan}\left(\color{blue}{\alpha}\right) \mathrm{tan}\left(\color{blue}{\beta}\right)}$$

https://www.alanshawn.com/latex3-tutorial/#latex3-regular-expression-xxviii

\regex_replace_all

https://tex.stackexchange.com/questions/422631/expl3-regex-and-missing-semicolon

\documentclass{article}

\usepackage{tikz,xparse}

\ExplSyntaxOn

\tl_new:N \l_bob_func_tl

\NewDocumentCommand \makelinesegment { m }
 {
  \pgfextra
  \tl_set:Nn \l_bob_func_tl { #1 }
  \regex_replace_all:nnN { ([0-9])- } { \1)--( } \l_bob_func_tl
  \regex_replace_all:nnN { : } { , } \l_bob_func_tl
  \exp_last_unbraced:NNV % this is not necessary, but I find it cleaner, correct me if I'm wrong
  \endpgfextra
  (\l_bob_func_tl)
 }

\ExplSyntaxOff

\begin{document}

% \makelinesegment{1:1--1:2-2:2} % do not use outside `\tikz` or `{tikzpicture}`

\tikz\draw\makelinesegment{1:1--1:2-2:2};

\end{document}

https://stackoverflow.com/questions/41655383/r-markdown-similar-feature-to-newcommand-in-latex

$\operatorname{Var}(X)$

$$\begin{aligned} \operatorname{Var}[Y] &= x \\ &= 3 \end{aligned}$$

$$0 = \frac{\partial}{\partial \color{blue} z_l}\big(\|h(z_{l-1})\cdot w_l- \color{blue} z_l\| + \lambda \| h( \color{blue} z_l)\cdot w_{l+1} - z_{l+1}\| \big)$$

https://tex.stackexchange.com/questions/353114/latex-equations-colour-all-instances-of-symbol

$$0 = \frac{\partial}{\partial{\ensuremath{\color{blue} z_l}}}\big(\|h(z_{l-1})\cdot w_l-{\ensuremath{\color{blue} z_l}}\| + \lambda \| h({\ensuremath{\color{blue} z_l}})\cdot w_{l+1} - z_{l+1}\| \big)$$

$$0 = \frac{\partial}{\partial{\color{blue} z_l}}\big(\|h(z_{l-1})\cdot w_l-{\color{blue} z_l}\| + \lambda \| h({\color{blue} z_l})\cdot w_{l+1} - z_{l+1}\| \big)$$

---

$$\require{color} \begin{aligned} 0\le\epsilon=\dfrac{\overline{PF}}{d\left(P,L\right)}=\dfrac{\overline{PF}}{\overline{PP^{\prime}}}= & \dfrac{\left\Vert \left(x,y\right)-\left(0,y_{{\scriptscriptstyle F}}\right)\right\Vert }{\left\Vert \left(x,y\right)-\left(x,y_{{\scriptscriptstyle L}}\right)\right\Vert }=\dfrac{\left\Vert \left(x,y-y_{{\scriptscriptstyle F}}\right)\right\Vert }{\left\Vert \left(0,y-y_{{\scriptscriptstyle L}}\right)\right\Vert }=\dfrac{\sqrt{x^{2}+\left(y-y_{{\scriptscriptstyle F}}\right)^{2}}}{\sqrt{\left(y-y_{{\scriptscriptstyle L}}\right)^{2}}}\\ \epsilon^{2}= & \dfrac{x^{2}+\left(y-y_{{\scriptscriptstyle F}}\right)^{2}}{\left(y-y_{{\scriptscriptstyle L}}\right)^{2}}=\dfrac{x^{2}+y^{2}-2y_{{\scriptscriptstyle F}}y+y_{{\scriptscriptstyle F}}^{2}}{y^{2}-2y_{{\scriptscriptstyle L}}y+y_{{\scriptscriptstyle L}}^{2}}\\ 0= & x^{2}+\left(1-\epsilon^{2}\right)y^{2}-2\left(y_{{\scriptscriptstyle F}}-\epsilon^{2}y_{{\scriptscriptstyle L}}\right)y+\left(y_{{\scriptscriptstyle F}}^{2}-\epsilon^{2}y_{{\scriptscriptstyle L}}^{2}\right)\\ \overset{\epsilon\ne1}{=} & x^{2}+\left(1-\epsilon^{2}\right)\left[y^{2}-\dfrac{2\left(y_{{\scriptscriptstyle F}}-\epsilon^{2}y_{{\scriptscriptstyle L}}\right)}{1-\epsilon^{2}}y+\dfrac{y_{{\scriptscriptstyle F}}^{2}-\epsilon^{2}y_{{\scriptscriptstyle L}}^{2}}{1-\epsilon^{2}}\right]\\ = & x^{2}+\left(1-\epsilon^{2}\right)\\ & \left[y^{2}-\dfrac{2\left(y_{{\scriptscriptstyle F}}-\epsilon^{2}y_{{\scriptscriptstyle L}}\right)}{1-\epsilon^{2}}y+\left(\dfrac{y_{{\scriptscriptstyle F}}-\epsilon^{2}y_{{\scriptscriptstyle L}}}{1-\epsilon^{2}}\right)^{2}-\left(\dfrac{y_{{\scriptscriptstyle F}}-\epsilon^{2}y_{{\scriptscriptstyle L}}}{1-\epsilon^{2}}\right)^{2}+\dfrac{y_{{\scriptscriptstyle F}}^{2}-\epsilon^{2}y_{{\scriptscriptstyle L}}^{2}}{1-\epsilon^{2}}\right]\\ = & x^{2}+\left(1-\epsilon^{2}\right)\left[\left(y-\dfrac{y_{{\scriptscriptstyle F}}-\epsilon^{2}y_{{\scriptscriptstyle L}}}{1-\epsilon^{2}}\right)^{2}+\dfrac{\left(y_{{\scriptscriptstyle F}}^{2}-\epsilon^{2}y_{{\scriptscriptstyle L}}^{2}\right)\left(1-\epsilon^{2}\right)-\left(y_{{\scriptscriptstyle F}}-\epsilon^{2}y_{{\scriptscriptstyle L}}\right)^{2}}{\left(1-\epsilon^{2}\right)^{2}}\right]\\ = & x^{2}+\left(1-\epsilon^{2}\right)\left(y-\dfrac{y_{{\scriptscriptstyle F}}-\epsilon^{2}y_{{\scriptscriptstyle L}}}{1-\epsilon^{2}}\right)^{2}+\dfrac{\left(y_{{\scriptscriptstyle F}}^{2}-\epsilon^{2}y_{{\scriptscriptstyle L}}^{2}\right)\left(1-\epsilon^{2}\right)-\left(y_{{\scriptscriptstyle F}}-\epsilon^{2}y_{{\scriptscriptstyle L}}\right)^{2}}{1-\epsilon^{2}} \end{aligned}$$

$$\require{color} \begin{aligned} & \left(y_{{\scriptscriptstyle F}}^{2}-\epsilon^{2}y_{{\scriptscriptstyle L}}^{2}\right)\left(1-\epsilon^{2}\right)-\left(y_{{\scriptscriptstyle F}}-\epsilon^{2}y_{{\scriptscriptstyle L}}\right)^{2}\\ = & \left(1-\epsilon^{2}\right)y_{{\scriptscriptstyle F}}^{2}-\left(\epsilon^{2}-\epsilon^{4}\right)y_{{\scriptscriptstyle L}}^{2}-y_{{\scriptscriptstyle F}}^{2}+2\epsilon^{2}y_{{\scriptscriptstyle F}}y_{{\scriptscriptstyle L}}-\epsilon^{4}y_{{\scriptscriptstyle L}}^{2}\\ = & -\epsilon^{2}y_{{\scriptscriptstyle F}}^{2}-\epsilon^{2}y_{{\scriptscriptstyle L}}^{2}+2\epsilon^{2}y_{{\scriptscriptstyle F}}y_{{\scriptscriptstyle L}}=-\epsilon^{2}\left(y_{{\scriptscriptstyle F}}-y_{{\scriptscriptstyle L}}\right)^{2} \end{aligned}$$

---

$$\require{color} \begin{aligned} 0\le\epsilon=\dfrac{\overline{PF}}{d\left(P,L\right)}=\dfrac{\overline{PF}}{\overline{PP^{\prime}}}= & \dfrac{\left\Vert \left(x,y\right)-\left(0,y_{{\scriptscriptstyle F}}\right)\right\Vert }{\left\Vert \left(x,y\right)-\left(x,y_{{\scriptscriptstyle L}}\right)\right\Vert }=\dfrac{\left\Vert \left(x,y-y_{{\scriptscriptstyle F}}\right)\right\Vert }{\left\Vert \left(0,y-y_{{\scriptscriptstyle L}}\right)\right\Vert }=\dfrac{\sqrt{x^{2}+\left(y-y_{{\scriptscriptstyle F}}\right)^{2}}}{\sqrt{\left(y-y_{{\scriptscriptstyle L}}\right)^{2}}}\\ \epsilon^{2}= & \dfrac{x^{2}+\left(y-y_{{\scriptscriptstyle F}}\right)^{2}}{\left(y-y_{{\scriptscriptstyle L}}\right)^{2}}=\dfrac{x^{2}+y^{2}-2y_{{\scriptscriptstyle F}}y+y_{{\scriptscriptstyle F}}^{2}}{y^{2}-2y_{{\scriptscriptstyle L}}y+y_{{\scriptscriptstyle L}}^{2}}\\ 0= & x^{2}+\left(1-\epsilon^{2}\right)y^{2}-2\left(y_{{\scriptscriptstyle F}}-\epsilon^{2}y_{{\scriptscriptstyle L}}\right)y+\left(y_{{\scriptscriptstyle F}}^{2}-\epsilon^{2}y_{{\scriptscriptstyle L}}^{2}\right)\\ \overset{\epsilon\ne1}{=} & x^{2}+\left(1-\epsilon^{2}\right)\left[y^{2}-\dfrac{2\left(y_{{\scriptscriptstyle F}}-\epsilon^{2}y_{{\scriptscriptstyle L}}\right)}{1-\epsilon^{2}}y+\dfrac{y_{{\scriptscriptstyle F}}^{2}-\epsilon^{2}y_{{\scriptscriptstyle L}}^{2}}{1-\epsilon^{2}}\right]\\ = & x^{2}+\left(1-\epsilon^{2}\right)\\ & \left[y^{2}-\dfrac{2\left(y_{{\scriptscriptstyle F}}-\epsilon^{2}y_{{\scriptscriptstyle L}}\right)}{1-\epsilon^{2}}y+\left(\dfrac{y_{{\scriptscriptstyle F}}-\epsilon^{2}y_{{\scriptscriptstyle L}}}{1-\epsilon^{2}}\right)^{2}-\left(\dfrac{y_{{\scriptscriptstyle F}}-\epsilon^{2}y_{{\scriptscriptstyle L}}}{1-\epsilon^{2}}\right)^{2}+\dfrac{y_{{\scriptscriptstyle F}}^{2}-\epsilon^{2}y_{{\scriptscriptstyle L}}^{2}}{1-\epsilon^{2}}\right]\\ = & x^{2}+\left(1-\epsilon^{2}\right)\left[\left(y-\dfrac{y_{{\scriptscriptstyle F}}-\epsilon^{2}y_{{\scriptscriptstyle L}}}{1-\epsilon^{2}}\right)^{2}+\dfrac{\colorbox{#FFFF66}{$\left(y_{{\scriptscriptstyle F}}^{2}-\epsilon^{2}y_{{\scriptscriptstyle L}}^{2}\right)\left(1-\epsilon^{2}\right)-\left(y_{{\scriptscriptstyle F}}-\epsilon^{2}y_{{\scriptscriptstyle L}}\right)^{2}$}}{\left(1-\epsilon^{2}\right)^{2}}\right]\\ = & x^{2}+\left(1-\epsilon^{2}\right)\left(y-\dfrac{y_{{\scriptscriptstyle F}}-\epsilon^{2}y_{{\scriptscriptstyle L}}}{1-\epsilon^{2}}\right)^{2}+\dfrac{\colorbox{#FFFF66}{$\left(y_{{\scriptscriptstyle F}}^{2}-\epsilon^{2}y_{{\scriptscriptstyle L}}^{2}\right)\left(1-\epsilon^{2}\right)-\left(y_{{\scriptscriptstyle F}}-\epsilon^{2}y_{{\scriptscriptstyle L}}\right)^{2}$}}{1-\epsilon^{2}} \end{aligned}$$

$$\require{color} \begin{aligned} & \colorbox{#FFFF66}{$\left(y_{{\scriptscriptstyle F}}^{2}-\epsilon^{2}y_{{\scriptscriptstyle L}}^{2}\right)\left(1-\epsilon^{2}\right)-\left(y_{{\scriptscriptstyle F}}-\epsilon^{2}y_{{\scriptscriptstyle L}}\right)^{2}$} \\ = & \left(1-\epsilon^{2}\right)y_{{\scriptscriptstyle F}}^{2}-\left(\epsilon^{2}-\epsilon^{4}\right)y_{{\scriptscriptstyle L}}^{2}-y_{{\scriptscriptstyle F}}^{2}+2\epsilon^{2}y_{{\scriptscriptstyle F}}y_{{\scriptscriptstyle L}}-\epsilon^{4}y_{{\scriptscriptstyle L}}^{2}\\ = & -\epsilon^{2}y_{{\scriptscriptstyle F}}^{2}-\epsilon^{2}y_{{\scriptscriptstyle L}}^{2}+2\epsilon^{2}y_{{\scriptscriptstyle F}}y_{{\scriptscriptstyle L}}=-\epsilon^{2}\left(y_{{\scriptscriptstyle F}}-y_{{\scriptscriptstyle L}}\right)^{2} \end{aligned}$$