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.

Existence of Four Triangle Centers

A vector proof

 Posted on September 18, 2022  |  4 minutes  |  Vincent Tam  |   0 comment

Settings Let $P$ be an arbitrary reference point. $\triangle ABC$ be a triangle. $\vec{a} = \overrightarrow{PA}$, $\vec{b} = \overrightarrow{PB}$, $\vec{c} = \overrightarrow{PC}$ Remark: $O$ is reserved for circumcenter. symbol name meaning $G$ centroid center of gravity $H$ orthocenter three “heights” are concurrent $I$ incenter center of inscribed circle $O$ circumcenter center of circumscribed circle Centroid Verify that $(\vec{a} + \vec{b} + \vec{c})/3$ satisfy the constraints. Orthocenter Let $H$ be the point of intersection of two altitudes $AA_H$ and $BB_H$ $\vec{h} = \overrightarrow{PH}$ \begin{align} (\vec{h} - \vec{a}) \cdot (\vec{b} - \vec{c}) &= 0 \\ (\vec{h} - \vec{b}) \cdot (\vec{c} - \vec{a}) &= 0 \end{align} Add these two equations together. [Read More]
geometry  vectors 

Katex Boxed Equations

 Posted on September 7, 2022  (Last modified on April 13, 2023)  |  1 minutes  |  Vincent Tam  |   0 comment

Goal To render a bordered box with multiple equations. Background My previous article about dot products contains some boxed math expressions: some are inline and some take up the whole line. For inline math, that’s simple: either $\fbox{text}$ or $\boxed{a=1}$ will do. For displayed math, these two KaTeX commands will ignore the newline command \\. Solution A simple Google search ‘“katex” AND “box”’ saved the day. It led me to this relevant Stack Overflow question, whose accepted answer suggested the use of custom CSS. [Read More]
KaTeX  CSS 

Tikz to SVG in VS Code

My little LaTeX Workshop recipe

 Posted on September 6, 2022  (Last modified on September 7, 2022)  |  2 minutes  |  Vincent Tam  |   1 comment

Goal View SVG graphics generated from TikZ code in VS Code. Compile TikZ code and preview SVG in VS Code with LaTeX Workship and SVG Preview Part I: generate SVG using LaTeX Workshop From the IDE’s parameters tab, I found the tools and recipes parameters. I clicked my user settings settings.json twice: once through the tools, and once through the recipes. Each click generated an array of default recipes or tools. [Read More]
VS Code  LaTeX Workshop 

Dot Products

My complicated derivation

 Posted on September 6, 2022  |  3 minutes  |  Vincent Tam  |   0 comment

Preface There’s a simpler derivation using the geometric defintion linearity from the definition 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: $\sum_{i=3}^6 i^2 = 3^2 + 4^2 + 5^2 + 6^2 = 86$ [Read More]
vectors  linear algebra 

Functions for Arts

Summary of my two-day work

 Posted on September 5, 2022  (Last modified on April 13, 2023)  |  6 minutes  |  Vincent Tam  |   0 comment

my HTML slides 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 minimize the calculations use daily-life examples favor pictures over text. IT skills used Inkscape C-S-a for alignment. C-S-f for boundary and colors, and line patterns (i.e. arrows). C-S-d for document size. clipping process: prepare the object to be clipped. [Read More]
functions  pandoc  CSS 

dvisvgm Guide for TikZ Pictures with Pattern Filling

A quick note of the right commands for the conversion

 Posted on June 21, 2022  |  2 minutes  |  Vincent Tam  |   0 comment

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

 Posted on June 20, 2022  |  4 minutes  |  Vincent Tam  |   0 comment

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

 Posted on June 18, 2022  (Last modified on June 19, 2022)  |  2 minutes  |  Vincent Tam  |   0 comment

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

 Posted on June 17, 2022  (Last modified on June 19, 2022)  |  2 minutes  |  Vincent Tam  |   0 comment

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

 Posted on June 15, 2022  |  1 minutes  |  Vincent Tam  |   0 comment

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