Sekai 🌐 πŸ—Ί

Sekai is the kanji for δΈ–η•Œ, meaning “the world”. That’s a great word because of the scale it designates.

dvisvgm Guide for TikZ Pictures with Pattern Filling

A quick note of the right commands for the conversion

Problem I had issues in getting a SVG from a TikZ picture containing a pattern filling: https://github.com/mgieseki/dvisvgm/issues/158. The reported issue was created last year, and it’s marked as fixed. Nonetheless, I had this issue even though my MikTeX was up-to-date. Discussion I wanted to confirm whether my dvisvgm.exe, which had been shipped with MikTeX, was latest. I first asked at https://github.com/mgieseki/dvisvgm/issues/184, but I was referred to MikTeX’s GitHub repository since it’s about the MikTeX-packaged version of this tool. [Read More]
LaTeX  dvisvgm  SVG 

Distance From a Line

TikZ figures for a derivation from dot product

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 $$ d((x_1,y_1), L) = \frac{\lvert Ax_1 + By_1 + C \rvert}{\sqrt{A^2 + B^2}}, $$ 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”. [Read More]
LaTeX 

Elementary Log Inequality

TikZ figures for some graphical proofs of this inequality

Goal Show that $$ \ln (x) \le x - 1 $$ 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 $$e^x \ge x + 1$$ for all $x \in \Bbb{R}$, which is obviously true. Proof by definite integrals Here’s a second proof using definite integrals. Case 1: x > 1 The rectangle $[1,x] \times [0,1]$ contains the region under the graph of the reciprocal $y = 1/x$ in the domain $[1,x]$. [Read More]
LaTeX 

A Plot for Euler's Constant

My first PGF Plot for step functions

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. [Read More]
LaTeX 

Simpler Diagram for Trigonometric Functions With Unit Circle

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)! [Read More]
LaTeX 

Tikz Illustration for Triangle Inequality

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. [Read More]
LaTeX 

Git Warning: LF Will Be Replaced by CRLF

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

LaTeX Table Background Color for TeXit

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). [Read More]
LaTeX  TeXit 

Line Equation in Intercept Form

LaTeX code for a TikZ figure

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. [Read More]
LaTeX 

LaTeX Multirow Multicol Table Coloring

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. [Read More]
LaTeX  table