Articles of euclidean geometry

Average Degree of a Random Geometric Graph

A set of $N$ points are distributed randomly on a unit square with uniform distribution. Two points $\mathbf{p}_i$ and $\mathbf{p}_j$ are said to be connected if $\|\mathbf{p}_i – \mathbf{p}_j\| \leq \delta$, where $0 < \delta < \sqrt2$. What is the expected average degree ($2M/N$) of this graph?

Finding the closest point in a set to another point in n-dimensional space: efficiently

I’m a programmer and am working on writing an efficient algorithm that, given a point P in n-dimensional space, can find the closest point from a set of points. For my problem, each of the n axis in the n-dimensional is constrained by a>=0 and a<=100, where a is the axis. There are n points […]

Why the number of symmetry lines is equal to the number of sides/vertices of a regular polygon?

Considering the/a definition of a regular polygon from Wiki : In Euclidean geometry, a regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length). , my question is how to prove that the number of symmetry lines is equal to the number of […]

What is the relationship between the parallel postulate and geodesic completeness?

Definitions: A geodesic space is a metric space $(X,d)$ such that every two points $x,y \in X$ can be joined by a geodesic. (A path $[0,1] \to X$ is a rectifiable curve if and only if it has a parametrization for which it is Lipschitz continuous. A rectifiable curve is a geodesic if and only […]

Relationship between the sides of inscribed polygons

In my math textbook there’s a demonstration for the calculus of the circumference of a circle that involves regular polygons inscribed in the circle, but I don’t get it. The book gives the following formula: $$l_{2n}=\sqrt{2R^2 – R\sqrt{4R^2-l_{n}^2}}$$ Where $l_{n}$ is the side of a regular n-sided polygon inscribed in a circle with R radius. […]

Cutting a square of area $A$ through It's mid point yields two polygons of area $A/2$ for arbitrary cuts?

A square of area $A$ is cut by a straight line at It’s mid point : It consists now of two rectangles, with area $A/2$. I want to find a proof that I could rotate the line and the area of the two polygons is still $A/2$, for example: And conclude that the area will […]

Points on two skew lines closest to one another

Given two skew lines defined by 2 points lying on them as $(\vec{x}_1,\vec{x}_2)$ and $(\vec{x}_3,\vec{x}_4)$. What are the vectors for the two points on the corrwsponding lines, distance between which is minimum? That distance is thus but what are the points where it is achieved?

Maximal area of a triangle

What would be the most elementary proof of the following: A triangle has been drawn inside the circle with radius $r$ and its area is as large as possible. Prove that the triangle is equilateral. I mean, can we avoid topology and compactness to prove that the maximal triangle can be found? And can we […]

Trying to understand the limit of regular polygons: circle vs apeirogon (vs infinigon?)

In the definition of regular polygon at the Wikipedia, there is this statement about the limit of a n-gon: “In the limit, a sequence of regular polygons with an increasing number of sides becomes a circle, if the perimeter is fixed, or a regular apeirogon, if the edge length is fixed.” My question is: I […]

Locus of intersection of two perpendicular normals to an ellipse

I came up with the following question while playing around with Geogebra, and rather than do it myself I figured I’d offer it here. For a given ellipse, the director circle is defined to be the set of all points where two perpendicular tangents to the ellipse cross each other. Suppose we instead consider the […]