Ulysses' trip

Latest post

Nov 19, 2024, 17:57:17

The <input type="number"> is so evil. I thought that entering 1e309 will have its value return 1e309 and that converting it to a number will give me infinity. Nope, it will return an empty string, and converting it to a number gets me zero. Note that 1e309 IS A VALID FLOATING-POINT NUMBER according to the #HTML standard.

#web #webdev #javascript #coding

Articles

Nov 12, 2022
Kinetic energy, momentum, and angular momentum of rigid bodies

In this article, we will find that the inertia matrix naturally appears when we calculate the kinetic energy $T$ or the angular momentum $\mathbf M$ of a rigid body. Then, we introduce the concept of principal inertia $\mathbf J_{\mathrm{pri}}$ . We also study how the inertia matrix changes under translations and rotations and how those transformations may lead to conclusions that can help us simplify the calculation of inertia matrices.
- Categories: physics
- Tags: rigid body, linear algebra, classical mechanics, from zhihu
Nov 11, 2022
Hölder means inequality

The Hölder mean of $\vec x$ with weights $\vec w$ and a parameter $p$ is defined as $M_{p,\vec w}\!\left(\vec x\right)\coloneqq\left(\vec w\cdot\vec x^p\right)^{\frac 1p}$ , and the value at $p=-\infty,0,+\infty$ are defined by the limits. We can prove using Jensen’s inquality that the Hölder mean increases as $p$ increases. This property can be used to prove HM-GM-AM-QM inequalities.
- Categories: math
- Tags: calculus, from zhihu
Nov 9, 2022
A polynomial whose sum of coefficients is a factorial

The function $\left(1-z\right)^{n+1}\sum_{k=1}^\infty k^nz^k$ is a polynomial of degree $n$ w.r.t. $z$ , and the sum of its coefficients is $n!$ . This turns out to be properties of Eulerian numbers.
- Categories: math
- Tags: combinatorics, number sequence, from zhihu
Nov 9, 2022
The image of a circular object through a thin lens

The image of a circle with radius $r$ and centered at $C\left(-2f,0\right)$ through a thin lens at $x=0$ with focal length $f$ and centered at $O\left(0,0\right)$ is a conic section with the focus being $\left(2f,0\right)$ , the directrix being line $x=f$ , and the eccentricity being $\frac rf$ .
- Categories: physics
- Tags: geometrical optics, from zhihu
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