Articles of geometry

Points in $ \mathbb{R}^{n} $

Let $ a_{1}, \dots a_{m} \in \mathbb{R}^{n} $ such that $ ||a_{i}-a_{j}||=1 \, \forall i \neq j $, where $ || \cdot || $ denotes the usual norm on $ \mathbb{R}^{n} $. Prove that $ m \leq n+1 $. I did it for the easy case when $ n=1 $ by explicit computation using the […]

Formula for cylinder

In an exercise I was asked to find a formula of the form $F(x,y,z)=C$ for a cylinder though the axis $(t,t,t)$ and radius $R$. The formula I got seemed a bit suspicious so I wanted to ask if I have it right. Basically I used the vector formula for the distance between a line and […]

Geometric Interpretation of Complex Algebraic Proof of Sum of Squares Statement

When I see answers regarding proofs such as the one mentioned here, it seems that there is a considerable diversity of ways to attempt to look at this proof. Similarly, although this sum of squares question is very different, I was wondering if there was a geometric interpretation to the following problem: Prove that given […]

coordinates of icosahedron vertices with variable radius

I was looking on the wikipedia page about icosahedrons and it says that for edge length $a$ the radius of the circumscribed sphere around the icosahedron is given by $r = a \times sin(\frac{2\pi}{5})$. It then says that the vertices of an icosahedron of edge length 2 are given by: $(0, ±1, ±\phi)$ $(±1, ±\phi, […]

Distances to line passing through the centroid of triangle

Let $p$ be a line that pass through the centroid of a triangle $ABC$. Unless the line pass through one vertex, then $2$ verices are one side of the line, while the third one is on the other side. Without loss of generality, let vertices $A$ and $C$ be one side, while the vertex $B$ […]

What is linear, numerically and geometrically speaking?

For as simple as it is, I never fully grasped what mathematicians and physicists mean with linear . Intuitively anything that looks like a straight line is interpreted as linear, like something in the form $f(x) = mx + q$ or any other function that maps $R \rightarrow R$ resulting in a graph that looks […]

How do you work with the space of circles on the sphere considered as the projective line?

I’m trying to prove some things about the action of the Möbius group on the “circlines” in the extended complex plane, ie. circles on $\mathbb{C}P^1$. I find that while I have a good grip on Möbius transformations (as $PSL(2, \mathbb{C})$), I don’t really know how I’m supposed to treat circles. Points on the sphere are […]

Determining the result of Boolean shape operations on closed Bézier shapes

Given two closed shapes made up of Bézier curves (and/or straight lines), I’m looking for an efficient way of calculating the resulting shape of the following Boolean operations: union difference intersection slice (imagine each of the shapes in the Venn diagram as its own shape; this operation is optional and can be expressed as a […]

The diagonals of a trapezoid are perpendicular and have lengths 8 and 10. Find the length of the median of the trapezoid.

The diagonals of a trapezoid are perpendicular and have lengths 8 and 10. Find the length of the median of the trapezoid. It this possible without a rhombus?

Curse of Dimensionality: hypercube inside a hypersphere

Exercise 2.5 of Izenman’s Modern Multivariate Statistical Techniques: Consider a hypercube of dimension $r$ and sides of length $2A$ and inscribe in it an $r$-dimensional sphere of radius $A$. Find the proportion of the volume of the hypercube that is inside the hypersphere, and show that the proportion tends to $0$ as the dimensionality $r$ […]