Sekai ๐ŸŒ ๐Ÿ—บ

Sekai (ไธ–็•Œ) is the kanji for “the world”. That’s a great word because of the scale that it designates.

Dot Products

My complicated derivation

Preface

There’s a simpler derivation using

  1. the geometric defintion
  2. linearity from the definition
  3. orthogonality of the canonical basic vectors $\vec{i}$, $\vec{j}$ and $\vec{k}$.

If I had read that, I wouldn’t have type this document in LaTeX. The following content was transcribed from a PDF file that I made three days ago.

Content

Recall: $\sum_{i=m}^{n} a_{i}=a_{m}+a_{m+1}+a_{m+2}+\cdots +a_{n-1}+a_{n},$ where $m$, $n$ are integers.

Example: $\sum_{i=3}^6 i^2 = 3^2 + 4^2 + 5^2 + 6^2 = 86$

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Functions for Arts

Summary of my two-day work

  1. my HTML slides
  2. You may view the source code of this blog to see the source files.

Goal

To provide an introduction to the formal definition of functions for arts stream students.

I aim to

  1. minimize the calculations
  2. use daily-life examples
  3. favor pictures over text.

IT skills used

Inkscape

  1. C-S-a for alignment.
  2. C-S-f for boundary and colors, and line patterns (i.e. arrows).
  3. C-S-d for document size.
  4. clipping process:
    1. prepare the object to be clipped.
    2. prepare the clip, which is on top of the previous object.
    3. select Object โ†’ Clip and then the first option from the top menu.

dvisvgm

Use the TEX โ†’ DVI โ†’ SVG route with latex and this tool with options --font-format=woff and -b 5 -R -d 2 to save file size and to ensure that the SVG can be correctly opened on Inkscape.

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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

  1. I wanted to confirm whether my dvisvgm.exe, which had been shipped with MikTeX, was latest.
  2. I gave a MWE illustrating the problem with the graph of y = (x + 1)ยฒ with region under the curve filled with a pattern in the above issue. I knew that’s probably off-topic, but the package owner might have some insights about the problem. It turned out that PDF โ†’ SVG was partly supported by dvisvgm. Using DVI โ†’ SVG, I managed to get the filling pattern, but I lost the beautiful Computer Modern Roman fontsโ€”that’s another issue.
  3. Hoping for a response from the owner for the previous problem, I started https://github.com/mgieseki/dvisvgm/issues/186. He suggested the use of --no-fonts or --font-format=woff. I observed no difference in the local filesystem. I uploaded everything I got through GitHub so that he could see what my dvisvgm.exe had output. I didn’t expected that the Computer Modern Roman fonts disappeared on GitHub side. If I hadn’t asked there, I would have never known that it’s the web browser’s security measure not to load the embedded font data in a SVG loaded from a server.

Skills learned

Meaning of some commonly used parameters of dvisvgm:

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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”.

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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.

PGF Plot for log inequality

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 $[1,x]$.

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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=1.3,
    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}
Euler's constant

Euler's constant illustration

SVG generated by dvisvgm

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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] ($(O)!0.3!(E)$) 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}

unit circle sine cosine

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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.

triangle inequality

\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.

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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

  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.

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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.

LaTeX table with TeXit problem

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