Articles of geometry

Infimum length of curves

Let the unit disc $\{(x,y): r^2=x^2+y^2<1\}\subset\mathbb R^2$ be equipped with the Riemannian metric $dx^2 +dy^2\over 1-(x^2+y^2)$. Why does it follow that the shortest/infimum length of curves are diameters? I remember doing an optimization course some time ago, but unfortunately all that was learnt has been unlearnt. Or perhaps there is a simpler way?

How to check does polygon with given sides' length exist?

I have polygon with $n$ angles. Then I have got $n$ values, which mean this polygon’s sides’ length. I have to check does this polygon exist (means – could be drawn with given sides’ length). Is there any overall formula to check that? (like e.g. $a+b\ge c$, $a+c\ge b$, $c+b\ge a$ for triangle)

Interpreting a geometry problem involving chords of a circle

The problem is stated as follows: Prove that if from the endpoints of a diameter of a circle, two intersecting chords are drawn, then the sum of the products of each chord and the segment of it from the endpoint of the diameter to the intersection point is a constant quantity. I do not want […]

Calculating centre of rotation given point coordinates at different positions

I am trying to work out if the centre of rotation of a measured sphere is actually at 0,0 or slightly offset from the centre. The situation is as follows: I have a machine tool with a table that rotates about its centre. I wish to verify that the centre of the table is at […]

Reference request: algebraic methods in geometry

I am a (soon to be) third year undergraduate who has just finished courses in linear and abstract algebra. While I enjoyed the study of algebraic structures in their own right, my favorite part of the courses were the applications of the algebraic machinery developed to geometric problems (i.e. the connection between Galois theory and […]

Problems with the definition of vectors as directed line segments in $\mathbb{R}^3$

First I’ll say where I’m working: The vectorial spaces $\mathbb{R}^2$ and $\mathbb{R}^3$. Then I’ll define a vector of this spaces as the following: $\textbf{Definition. }$ A vector $\vec{v}$ is the set of all equal directed line segments. Now suppose that $$\underbrace{\overrightarrow{AB}}_{\mbox{directed line segment}} \in \vec{v},$$ which is a correct notation, by definition. So why do […]

How to fill up the gap between a typical advanced undergraduate algebraic curve course and High school basic geometry/precalculus course?

Based on this question i asked recently: A question about geometry of plane curve books, i think it is too advance for a HS student/ typical second or third year undergraduate math majors to read on their own on the books given on the answer to that question Also, i think it is too much […]

Predicting Spirals

I am currently in the process of analyzing a polyspiral, a spiral where each successive length drawn is increased at specified increment at the same angle. *Please note the angles selected are the exterior angles or the angle by how much the turtle turns by. 144 degrees: 216 degrees: versus 140 degrees or 120 degrees: […]

Nearest point on the line or its extension from a point

What is easiest way to find the a point on a line $(a1, b1)$, $(a2, b2)$ or the extension of the line, which is nearest to a point $(x1, y1)$.

Circumradius of a hexagon with side lengths $2$, $2$, $7$, $7$, $11$, $11$

This question already has an answer here: Cyclic Hexagon Circumradius 3 answers