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• Nov 8, 2022

  The point on the circle farthest to two lines

  Suppose $P$ is a point on the circle $\odot C$. When is the sum of distances from $P$ to two edges of $\angle O$ extremal? It turns out to be related to angle bisectors (the intersections of $\odot C$ and the bisector of $\angle O$ or its adjacent supplementary angle are extremals), while the edge cases (at the intersections of $\odot C$ and edges of $\angle O$) are a little tricky: we need to use the bisectors to divide the plane into four quadrants, pick the two quadrants where the line intersecting $\odot C$ at $P$ lies, translate the region to make it center at $C$, and see whether $P$ is inside the translated region.

  • Categories: math
  • Tags: elementary geometry, trigonometry, from zhihu
• Nov 8, 2022

  Typesetting math in emails

  After some comparison among solutions, I use KaTeX to typeset math in my emails.

  • Categories: programming
  • Tags: jekyll, ruby, tex
• Nov 7, 2022

  Mapping from Kepler problem to free particle on 3-sphere

  There is a canonical transform of the Kepler problem which is the same as the problem of motion of a free particle on 3-sphere. The explicit formula of the transform as well as some links about this topic is written in the article. The explicit formula for $E<0$ is $\mathbf u\coloneqq\frac{p^2-p_0^2}{p^2+p_0^2}\hat{\mathbf n}+\frac{2p_0}{p^2+p_0^2}\mathbf p,$ where $\mathbf u$ is the position of the particle on 3-sphere (a 4-dimensional vector), $\mathbf p$ is the momentum of the original particle in Kepler problem, $\hat{\mathbf n}$ is a vector perpendicular to the 3-dimensional hyperplane where $\mathbf p$ lies, and $p_0\coloneqq\sqrt{-2mE}$.

  • Categories: physics
  • Tags: classical mechanics, canonical transformation, letter, kepler problem
• Nov 6, 2022

  Solving linear homogeneous ODE with constant coefficients

  By using power series, we can prove that the problem of solving linear homogeneous ODE with constant coefficients can be reduced to the problem of solving a polynomial with those coefficients. This article illustrates this point in detail, but it uses a very awful notation…

  • Categories: math
  • Tags: calculus, linear algebra, combinatorics, ode, long paper, from zhihu
• Nov 6, 2022

  Thoughts on a middle school thermal physics problem

  To heat an object with hot water, if we divide the water into more parts and use each part to heat the object one after another, the final temperature will be higher. If the number of parts tends to infinity, then the final temperature will tend to the limit $T+\mathrm e^{-\frac C{C_0}}\left(T_0-T\right)$, where $T$ is the initial temperature of the object, $T_0$ is the temperature of the hot water, $C$ is the heat capacity of the object, and $C_0$ is the heat capacity of the hot water.

  • Categories: physics
  • Tags: calculus, number sequence, from zhihu
• May 23, 2022

  I created a community rhythm game called Dododo

  I have created a community rhythm game called Dododo. It is a rhythm game with musical rhythm notations.

  • Categories: update
  • Tags: update, rhythm game, web
• Aug 21, 2020

  New website for MOIS Project (club)

  I have built a website for our club called MOIS Project. It is a club related to music and band.

  • Categories: update
  • Tags: update
• Jul 28, 2020

  My activities in 2020 JDFZ Summer Session

  JDFZ is hosting a summer session, where lectures covering a wide range of topics are held. Come and see what are my activities during this event.

  • Categories: update
  • Tags: update
• Jul 5, 2020

  New website for Little Turings (club)

  I, the president and one of the founders of Little Turings, created a website to distribute information about it. The website is bilingual (Chinese and English).

  • Categories: update
  • Tags: update
• Jun 13, 2020

  Drawing a heart using Joukowsky transformation

  Joukowsky transformation of a circle centered at $\left(1,1\right)$ of radius $1$ is a curve resembling a heart.

  • Categories: math
  • Tags: complex

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