Articles of geometry

How to calculate the area of bizarre shapes

I’m looking for an algorithm to calculate the area of various shapes (created out of basic shapes such as circles, rectangles, etc…). There are various possibilities such as the area of 2 circles, 1 triangular and 1 square (intersections possible). As you can see this gets quite complicated and requires a numeric solution. Of course, […]

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?