Consider the unit-ball in Dimension $\mathbb{R}^d$.
Say I know two normalized vectors (thus 'pointing' on two points $p$ and $q$ on the unit sphere) such that the angle between them is either $\alpha$ or $\beta$.
From this I want to calculate the euclidean distance between the two points $p$ and $q$. It is clear that there are only two-possible distances and that the distance heavily depends on $\alpha$ and/or $\beta$.
I believe that there already is some formula for this but I didn't find any, neither i came up with one.
I added an example with in dimension $2$ and angle $\alpha$ for reference. The distance I want to find is the blue line between $p$ and $q$ for known $\alpha$.
