Subtended Angles

Abstract

We consider the following question. Suppose that $d\ge 2$ 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 $dm-\left(\genfrac{}{}{0ex}{}{d+1}{2}\right)-1$ 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.