Articles of geometry

Converting a rotated ellipse in parametric form to cartesian form

I have a rotated ellipse in parametric form: $$\begin{pmatrix}y \\ z\end{pmatrix} = \begin{pmatrix}a\cos t + b\sin t \\ c\cos t + d\sin t\end{pmatrix} \tag{1} $$ or, $$(y,z) = (a\cos t + b\sin t , c\cos t + d\sin t) \tag{2} $$ By using $$\cos^2 t + \sin^2 t = 1 $$ I can rewrite into: […]

Examples of interesting / non-trivial manifolds that are direct products

What are interesting / non-trivial examples of smooth connected closed manifolds that are direct products or involve direct products? I am especially interested in orientable manifolds. Say, an $n$-torus $T^n$ is a direct product of $n$ copies of a circumference $S^1$. One can build a 3-manifold from a surface of genus $g$ as $M=M^2_g\times S^1$, […]

The origin is not in the convex hull $\Rightarrow$ the set lies in a hemisphere?

I am trying to understand the proof of the following claim: Let $f:A \subseteq \mathbb{S}^n \to \mathbb{S}^n$ be an $L$-Lipschitz* map (with $L <1$). Then $f(A)$ is contained in the interior of a hemisphere. *The distance on $\mathbb{S}^n$ can be either the intrinsic one or the extrinsic (Euclidean) one, it does not matter. In the […]

Geodesic equations and christoffel symbols

I want to learn explicitly proof of the proposition 9.2.3. Which books or lecture notes I can find? Please give me a suggestion. Thank you:)

How to make 3D object smooth?

I want to make the below picture into an egg with smooth surface. For the implementation in Mathematica, please, see this thread here. This thread considers mathematical methods to achieve the goal while the last one only in a single program. How can you make the 3D object smooth mathematically?

What is a straight line?

I have researched this question for days and can not locate a good answer. It could be a mathematical object that is defined by an axiom as Euclid or Hilbert. But if a curve is drawn between two points can it be should using only the rules of plane geometry that the curve is a […]

Three tangent circles inside a larger circle

Suppose you’re given a circle with center $O$, I’m curious, how can one construct with ruler and compass three circles inside the larger circle such that each is tangent to the larger circle as well as to the other two?

Is there an equidissection of a unit square involving irrational coordinates?

An equidissection of a square is a dissection into non-overlapping triangles of equal area. Monsky’s theorem from 1970 states that if a square is equidissected into $n$ triangles, then $n$ is even. In 1968, John Thomas proved the following weaker statement: there is no equidissection of a unit square into an odd number of triangles […]

Find the coordinates in an isosceles triangle

Given: $A = (0,0)$ $B = (0,-10)$ $AB = AC$ Using the angle between $AB$ and $AC$, how are the coordinates at C calculated?

Center of mass of an $n$-hemisphere

Related to this question. Note that I’m using the geometer definition of an $n$-sphere of radius $r$, i.e.$ \\{ x \in \mathbb{R}^n : \|x\|_2 = r \\} $ Suppose I have an $n$-sphere centered at $\bf 0$ in $\mathbb{R}^n$ with radius $r$ which has been divided into $2^k$ orthants by $k$ axis-aligned hyperplanes (note, $k […]