Background

A secondary school student posted a coordinate geometry question on a Discord homework help server. A helper replied to that question with an illustration of the Perpendicular Distance Formula

where $L$ is the line $Ax + By + C = 0$ without proof.

Goal

To provide an illustrated derivation of this formula.

Recall

A basic property of the dot product: the “algebraic definition” is equivalent to the “geometric definition”.

Goal

Show that

for all $x > 0$.

Proof by inverse functions

Think about their reflection along the line $y = x$ (i.e. their inverse function). Then we get

for all $x \in \Bbb{R}$, which is obviously true.

Proof by definite integrals

Here’s a second proof using definite integrals.

The rectangle $[1,x] \times [0,1]$ contains the region under the graph of the reciprocal $y = 1/x$ in the domain $[1,x]$.

I firsted tried with foreach, but that would create so much paths. I found them difficult to operate on later, for example, with PGF plots library fillbetween. const plot is a better solution.

\documentclass{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}
\usetikzlibrary{patterns}
\usepgfplotslibrary{fillbetween}
\begin{document}
\newcommand\myN{8}
\pgfplotsset{
    axis lines=center,
    legend style={at={(1,1)},anchor=north east,fill=none},
    title style={at={(0.5,1.05)}},
    every axis x label/.style={
        at={(ticklabel* cs:1)},
        anchor=west,
    },
    every axis y label/.style={
        at={(ticklabel* cs:1)},
        anchor=south,
    },
}
\begin{tikzpicture}
\begin{axis}[
    title={sum of hatched region converges to Euler's constant},
    xlabel={$x$},
    ylabel={$y$},
    xmin=0,
    xmax={\the\numexpr\myN+2},
    ymin=0,
    ymax=1.3,
    xtick=\empty,
    ytick=\empty,
    extra x ticks={1, \myN},
    extra x tick labels={$1$, $n$},
    extra y ticks={1},
]
\addplot[name path=A,domain=1:\myN,samples=501,smooth] {1/x} \closedcycle;
\addlegendentry{$y = \frac{1}{x}$};
\addplot+[domain=1:\myN+1,samples=\myN+1,jump mark left,blue,mark=*,mark options={draw=blue,fill=blue}] {1/x};
\addlegendentry{$y = \frac{1}{\lfloor x \rfloor}$};
\addplot+[name path=B,domain=1:\myN+1,samples=\myN+1,jump mark left,mark=none, draw=none] {1/x} \closedcycle;
\addplot[pattern=north east lines] fill between [of=A and B];
\end{axis}
\end{tikzpicture}
\end{document}

Background

In my previous post about unit circle and trigonometric functions, I included a graph with three trigonometry functions. I’m quite satisfied with my TikZ picture.

Problem

Unluckily, a secondary school student found that my diagram was too complicated.

Solution

\documentclass[tikz,border=2pt]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}
\usetikzlibrary{calc}
\newcommand\mytheta{110} %angle theta
\begin{document}
\begin{tikzpicture}[scale=2]
\coordinate[label=left:$O$] (O) at (0,0);
\coordinate[label=above left:${A = (\cos\theta, \sin\theta)}$] (A) at (\mytheta:1);
\coordinate[label=below left:${B = (\cos(-\theta), \sin(-\theta))}$] (B) at (-\mytheta:1);
\coordinate[label=right:$E$] (E) at (1,0);
\draw (O) circle (1);
\draw (A) -- (O)  node [midway, left] {$1$}
    -- (E);
\draw (B) -- (O);
\draw[-stealth] ($(O)!0.3!(E)$) arc (0:\mytheta:0.3) node[midway, above] {$\theta$};
\draw[-stealth] ($(O)!0.25!(E)$) arc (0:{-\mytheta}:0.25) node[midway, below] {$-\theta$};
\end{tikzpicture}
\end{document}

I thought the code would be easy, but it turned out that I spent one hour and half to draw this figure.

\documentclass[border=2pt,preview]{standalone}
\usepackage{tikz}
\usepackage{caption}
\newcommand{\sideA}{2}
\newcommand{\sideB}{3}
\newcommand{\sideC}{6}
\colorlet{myred}{red!40!yellow}
\colorlet{mygreen}{green!40}
\colorlet{myblue}{blue!20}
\begin{document}
\begin{figure}
\centering
\begin{tikzpicture}[thick]
\coordinate (A) at (0,0);
\coordinate (B) at (\sideC,0);
\draw[<->,>=stealth,myred] (A) -- ++ (-\sideA, 0) node [pos=0.5,below] {$\color{myred}{a}$};
\draw[<->,>=stealth,mygreen] (B) -- ++ (\sideB, 0) node [pos=0.5,below] {$\color{mygreen}{b}$};
\draw[<->,>=stealth,myblue] (A) --  (B) node [pos = 0.5, below]{$\color{myblue}{c}$};
\draw[dashed,myred] (A) circle (\sideA);
\draw[dashed,mygreen] (B) circle (\sideB);
\end{tikzpicture}
\captionsetup{labelformat=empty}
\caption{triangle inequality ${\color{myred}{a}} + {\color{mygreen}{b}} >
  {\color{myblue}{c}}$ with $\color{myred}{a = \sideA}$,
  $\color{mygreen}{b = \sideB}$, $\color{myblue}{c = \sideC}$}
\end{figure}
\end{document}

The colors are customized for the Discord bot TeXit.

I’m using Git Bash while writing this post. I’ve never tested the commands on *nix.

Background

Some of the tags and titles were written as “LateX”/“latex” in the source files. I batch corrected them with Git Grep and GNU Sed.

git grep --name-only -IiE ^-\ latex$'\r'? content/{post,page} | \
xargs -n1 sed -i "s/^- latex/- LaTeX/I"

I tried to match

1. a leading - using ^-
2. a whitespace \ escaped by a backslash to avoid wrapping the special character below with double quotes
3. the string “latex” (case-insensitive)
4. the carriage return \r, which is represented by $'\r' in bash, for at most once (?)

I observed that wrapping $'\r' with "" would lead to no match.

Problem

In my previous post about LaTeX tables, I used \cellcolor{white} to override the \rowcolor{cyan} in a cell in the leftmost column, which was the column type name.

On Discord, that cell became black because TeXit inverted black and white.

Discussion

I first tried blending black and white, and I found that black!27 and black!28 give a good approximation to Discord’s background color.

Thanks to this Discord Color Palette and a bot command channel on a server, I’ve observed that the desired color is RGB(54,57,62).

This is a first draft of a TikZ picture illustraing this classical formula to be used for math help channels.

Adding \caption{for the picture} without “Figure 1:” requires \usepackage{caption} and wrapping with \begin{figure}. It also possible to use the primitive TeX command \par, but it would be complicated to use that with standalone. In the previous post, the SVG picture from the LaTeX table in an article has too much useless whitespace around the table. I don’t bother to learn other packages, as I need time for other more important stuff.

Inspiration

Number of two-digit numbers not containing the digit ‘8’.

Code

• The standalone class can’t be used with \caption.
• *{8}{c} means 8 c.
• >{\columncolor{cyan}}c gives the column color. It \usepackage[table]{xcolor}.
• \rowcolor{cyan} overrides the column colors.
• \cellcolor{white} overrides the above two commands. It has to be carried out one-by-one without any fancy package.
• \multicolumn{n}{<alignment>}{content} defines a multi-column cell.
• \multirow{n}{*}{content} defines a multi-row cell. It \usepackage{multirow}. It can’t contain \multicolumn.
• \cline{i-j} draws a horizontal line above the row from column i to column j.
• No fancy way of coloring multirow cells without fancy packages.
• \multirow{-9} at the last row creates a cell spanning nine rows from bottom to above, so that its content won’t be covered by colors of former rows.
• \rotatebox[origin=[clr]]{angle}{content}: rotate content by angle anticlockwise with [clr] as the rotation center.

\documentclass[12pt]{article}
\usepackage{multirow}
\usepackage[table]{xcolor}
\usepackage{tikz}
\begin{document}
\begin{table}[htbp]
\centering
\caption{Sample \LaTeX{} multirow table}
\vspace*{1ex}
\label{multiprogram}
\begin{tabular}{|c|c|*{8}{c}>{\columncolor{cyan}}cc|}
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{}} & \multicolumn{10}{c|}{unit digit} \\\cline{3-12}
\multicolumn{2}{|c|}{} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\\hline
& 1 & &  &  &  &  &  &  &  &  &  \\
& 2 & &  &  &  &  &  &  &  &  &  \\
& 3 & &  &  &  &  &  &  &  &  &  \\
& 4 & &  &  &  &  &  &  &  &  &  \\
& 5 & &  &  &  &  &  &  &  &  &  \\
& 6 & &  &  &  &  &  &  &  &  &  \\
& 7 & &  &  &  &  &  &  &  &  &  \\
\rowcolor{cyan} \cellcolor{white}
& 8 & &  &  &  &  &  &  &  &  &  \\
\multirow{-9}{*}{\rotatebox[origin=c]{90}{tens digit}}
& 9 & &  &  &  &  &  &  &  &  &  \\\hline
\end{tabular}
\end{table}
\end{document}

Result:

Background

Beautiful Hugo comes with Font Awesome icons. However, it seems that it’s no longer maintained.

Goal

To upgrade Font Awesome to the current stable version (6.1.1).

Problem

The official guide for self-hosting Font Awesome Web Fonts + CSS suggests users to place the following files

into somewhere in my theme. The left column of the diagram features the folder assets/.