Articles of geometry

Voronoi diagram with different metric functions

Given a metric space $(X,d)$ and finite number of points $(x_i)_{i=1}^n$ the Voronoi diagram (or the Dirichlet cell) $C_i$ is given by $$ C_i = \{x\in X:d(x,x_i)<\min\limits_{j\neq i}d(x,x_j)\}. $$ I have two question: Does anybody know if it was considered the case with different distance function, namely $$ C'_i = \{x\in X:d_i(x,x_i)<\min\limits_{j\neq i}d_j(x,x_j)\}. $$ where […]

What do you call 'perpendicular but skew' lines?

For example, the seat tube and rear axle of a bicycle or motorcycle. That is, when viewed from above, the seat tube would appear ‘perpendicular’ to the rear axle. But in 3d reality, the lines are skew. I was wondering if there is a single word or concise way to describe such lines.

Prove that with 2 parallel planes, the one in between is given by

I would like to prove that having two planes $$ax+by+cz+d_1 = 0 \quad\text{and}\quad ax+by+cz+d_2 = 0$$ you can automatically have a plane with equal distance from each plane that looks like this: $$ax+by+cz+\frac{d_1+d_2}{2}=0.$$ I have tried deriving this from the formula for the distance of a point to a plane, but with no success. Any […]

Prove that $SC=SP$ if and only if $MK=ML.$

Point $P$ lies inside triangle $ABC$. Lines $AP$, $BP$, $CP$ meet the circumcircle of $ABC$. again at points $K, L, M,$ respectively. The tangent to the circumcircle at $C$ meets line $AB$ at $S$. Prove that $SC=SP$ if and only if $MK=ML.$ any help?

Calculating Solid angle for a sphere in space

How can I calculate the solid angle that a sphere of radius R subtends at a point P? I would expect the result to be a function of the radius and the distance (which I’ll call d) between the center of the sphere and P. I would also expect this angle to be 4π when […]

Classical Impossible Constructions of Geometry proofs with Abstract Algebra.

Trisecting an angle (dividing a given angle into three equal angles), Squaring a circle (constructing a square with the same area as a given circle), and Doubling a cube (constructing a cube with twice the volume of a given cube). Told that these problems could only be proved with abstract algebra. I have no idea […]

bounding the sum of squares of lengths of a quadrilateral inscribed in a unit square

Consider this nice little problem: if $ABCD$ is a quadrilateral inscribed in a unit square, then $$2\leq AB^2+BC^2+CD^2+DA^2\leq4$$ (Evidently this is problem 1 on paper 1 of the 1989 Irish Mathematical Olympiad. I also found it in another nice collection of problems, a recent admissions exam for the Indian Statistical Institute.) Here’s the straightforward algebraic […]

Finding the location of the end of an arc, knowing the beginning, the arc's length and the radius

I apologise in advance if this is really basic. I have a circle of radius $15$, from which I work out an arc, given an angle of arbitrary value (it’s for a computer program). Given that I know the point where the arc starts, how would I go about working out the coordinates of the […]

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)