Subtended Angles

TL;DR

This paper investigates the minimal number of points needed to realize given angles in Euclidean spaces of various dimensions, revealing surprising limitations and bounds on realizability.

Contribution

It establishes sharp bounds for angle realizability in 2D and shows these bounds do not extend to higher dimensions, providing new insights into geometric configurations.

Findings

Maximum realizable angles in 2D is 2m-4 for m points.

Bound of 2m-4 angles cannot be improved in 2D.

Existence of 2m-3 angles that cannot be realized by m points in any dimension.

Abstract

We consider the following question. Suppose that $d\ge2$ and $n$ are fixed, and that $\theta_1,\theta_2,\dots,\theta_n$ are $n$ specified angles. How many points do we need to place in $\mathbb{R}^d$ to realise all of these angles? A simple degrees of freedom argument shows that $m$ points in $\mathbb{R}^2$ cannot realise more than $2m-4$ general angles. We give a construction to show that this bound is sharp when $m\ge 5$. In $d$ dimensions the degrees of freedom argument gives an upper bound of $dm-\binom{d+1}{2}-1$ general angles. However, the above result does not generalise to this case; surprisingly, the bound of $2m-4$ from two dimensions cannot be improved at all. Indeed, our main result is that there are sets of $2m-3$ of angles that cannot be realised by $m$ points in any dimension.