# Posts

## The Bézout-Theorem and Pascal's Hexagon

This posting contains large animations.

## HEVC-Encoding

We’re going to encode a video to HEVC, and we basically have two choices what hardware to use for that task: CPU or GPU.

## The Viviani-Body

Povray in action.

## Intersecting Circles IV

We’re going to construct a line tangential to two given circles.

## Intersecting Circles III

Given a square and an arbitrary length $l$, we’re going to construct a rectangle with one of its edges being of length $l$ such that the areas of the square and the rectangle coincide. In the following sketch, two circles with same radius $\overline{AC}$ intersect in such a manner that their respective centers are located on the other circle’s boundary.

## Triangles I

For a triangle $\bigtriangleup_{ABC}$ its orthic is defined to be the triangle created by the foot of the altitudes dropped from the vertices of $\bigtriangleup_{ABC}$. We will see that under all inscribed triangles, the orthic has minimum perimeter. Of course, the triangle needs to be acute, as for obtuse triangles, the orthocenter lies outside $\bigtriangleup_{ABC}$, and for a right triangle, the orthocenter coincides with either $A$, $B$, or $C$.

## Intersecting Circles II

The app Euclidea, which I have mentioned in the last post, has an interesting level 4.2. I say interesting, because I was close to beaking my head while trying to solve it for the first time, although its solution is surprisingly easy. Suppose that we’re given a straight line and an external point $P$. The exercise is to construct a second line through $P$ which cuts the given one at $\delta = \displaystyle\frac{\pi}{3} = 60^°$.

## Intersecting Circles I

Let there be given a pasture with circular shape and diameter $r$ and let a sheep be attached to a point on the perimeter by a rope of length $l$. Then, how long does $l$ need to be so that the sheep can reach exactly half the pasture? The answer is $$\frac{l}{r} = {\verb+1.15872847301812+}$$ and the value can only be achieved by numerical means. Everything needed to derive this result is depicted and named in the following sketch.

## Very Simple Maths

Let’s do some very simple maths. Assume that we are given a closed curve $\gamma \subset \mathbb{R}^2$, parametrised via its arc-length $r \in [0,s]$, and two functions $f$ and $g$ defined there. We make the additional assumption that

$$\int_0^s f( r ) dr = 0$$

and we thereby express that the mean value of $f$ vanishes. We set

## Mail::GnuPG.pm: Encrypting an already signed email

There’s a problem with Mail::GnuPG.pm for which I have filed a bug-request: