Articles of spherical trigonometry

Random points on a sphere — expected angular distance

Suppose we randomly select $n>1$ points on a sphere (all independent and uniformly distributed). What is the expected angular distance from a point to its closest neighbor? What is the expected angular distance from a point to its $m^{\text{th}}$ closest neighbor (where $m<n$)?

Identity between $x=y+z$ and $\tan\left(\frac{\theta}{2}\right)=\tan\left(\frac{\nu}{2}\right)\tan\left(\frac{\pi/2-\epsilon}{2}\right) $

I would like to prove that (1) $$\begin{equation} \tan\left(\frac{\theta}{2}\right)=\tan\left(\frac{\nu}{2}\right)\tan\left(\frac{\pi/2-\epsilon}{2}\right) \end{equation}$$ can transformed to (2) $$x=y+z,$$ where (3) \begin{align} x&=&\mathrm{arctanh}\left(cos(\theta)\right)\\y&=&\mathrm{arctanh}\left(cos(\nu)\right)\\z&=&\mathrm{arctanh}\left(\sin\left(\epsilon\right)\right) \end{align} By solving for $\theta$ in 1 and 2, we see that these are indeed equal: For the record, incorrect identity Initially the question was wrongly stated, and the comments below pertain to this: I would […]

Combinatorics for a 3-d rotating automaton

Let’s suppose that we have some kind of special 3-dimensional rotating automaton. The automaton is capable to generate rotation about selected $X$ or $Y$ or $Z$ axis (in a current frame) in steps by only constant +$\dfrac{\pi}{6}$ angle (i.e. rotation can be generated only in one direction – reverse rotation is prohibited) so transition from […]

Step forward, turn left, step forward, turn left … where do you end up?

Take $1$ step forward, turn $90$ degrees to the left, take $1$ step forward, turn $90$ degrees to the left … and keep going, alternating a step forward and a $90$-degree turn to the left. Where do you end up walking? It’s very easy to see that you end up walking on the perimeter of […]