Articles of geometry

How to transform (rotate) this hyperbola?

Given this hyperbola $x_1^2-x_2^2=1$, how do I transform it into $y_1y_2=1$? When I factor the first equation I get $(x_1+x_2)(x_1-x_2)=1$, so I thought $y_1=(x_1+x_2)$ and $y_2=(x_1-x_2)$. Therefore the matrix must be $ \begin{pmatrix} 1&1\\1&-1 \end{pmatrix} $. But in maple it is just drawn as a straight line. However, it should be a rotated hyperbola. Is […]

Is it possible to create a volumetric object which has a circle, a square and an equilateral triangle as orthogonal profiles?

This question was posed to me by a friend (formulated as creating a peg to fit perfectly into holes of these shapes), and after an experiment in OpenSCAD it seems it is not possible – either one profile has to be an isosceles triangle rather than equilateral, rectangular rather than square, or elliptical rather than […]

Möbius map from circles to lines

I want to find a Möbius transformation that takes the two circles $C(i,3), C(-i,1)$ to the parallel lines $Re(z)=0, Re(z)=1$. I know that they intersect at -2i, which means I have to map $2i \mapsto \infty$. I’m not sure how to ensure that both get mapped to the appropriate line. I’ve tried using 2 other […]

Is there a Möbius transformation that scales disks to the unit disk?

When working in the complex plane, often times I would like to scale a disk $|z-z_0|<R$ to the unit disk. I would first translate $z_0$ to the origin, but after that, what can we multiply by to scale the radius down from $R$ to $1$? I’m curious because in reading a proof of Schwarz’ lemma, […]

Maximum number of acute triangles

Given $n$ points on the plane, no three of which are collinear, what is the maximum number of acute triangles formed by them? [Source: Based on Hungarian competition problem]

The injectivity of torus in the category of abelian Lie groups

HI: I have the following question: Definition: A Lie group $T$ is called a torus if $T\cong \prod_{1\leq i\leq k} \mathbb{R}/\mathbb{Z}$ for some $k\in \mathbb{N}$. ${\bf Question}$: Is it true that a torus is an injective object in the category of abelian Lie groups? Thanks very much!

Find out minimize volume (V) of tetrahedral

I have this problem: On space $ (Oxyz)$ given point $M(1,2,3)$. Plane ($\alpha$) contain point $M$ and ($\alpha$) cross $Ox$ at $A(a,0,0)$; $Oy$ at $B(0,b,0)$; $C(0,0,c)$. Where a,b,c>0 Write the equation of plane ($\alpha$) such that It makes $V_{OABC}$ reach minimum. I don’t know which inequality should use in here to find out $\min_{V_{OAB}}$ . […]

Umbilic points on a connected smooth surface problem

Let $S\subset \mathbb{R}^3$ be a connected smooth surface. Suppose that every point of $S$ is an umbilic point. Prove that $S$ is a subset of either a plane or a sphere in $\mathbb{R}^3$. Here’s a HW problem. I wonder how to prove it.

How do I determine if a point is within a rhombus?

I know the coordinates of the 4 rhombus’ vertices. I also have the coordinates of another arbitrary point (the result of a click on the screen). How do I determine if that point is within the rhombus?

Plane intersecting line segment

I have a plane which is represented as a 3d point $\vec{p}$ with a normal $\hat{n}$. I also have a line segment specified by two points $\vec{v_1},\vec{v_2}$ . I want to get the intersection point (if any). Here’s what I have: $${\rm dist}_{v_1} = \hat{n} \cdot (\vec{v_1} – \vec{p})$$ $${\rm dist}_{v_2} = \hat{n} \cdot (\vec{v_2} […]