Context: I'm currently looking at the dynamics of powers of a real symmetric matrix $M$. For a vector $\vec{v}\in \mathbb{R}^n$, I've found it useful to consider how $M^k\vec{v}$ `tends to' some ray from the origin (in the sense that the angle between $M^k\vec{v}$ and the ray goes to $0$ as $k$ goes to infinity).
By a ray from the origin in $\mathbb{R}^n$, I mean a set of the form $\{t\vec{v} : t> 0\}$, where $\vec{v}$ is a fixed non-zero vector. Let $\mathcal{R}^n$ be the set of all these rays. We can easily find a nice metric for $\mathcal{R}^n$ by considering the angle between two rays, or the euclidean distance between the points where two rays intersect the unit sphere.
Question: is there a common name for the space $\mathcal{R}^n$ with either of the above metrics? It's clear that our space is isometric with the sphere $S^{n-1}$, but it'd be much more convenient for me to work with the rays as rays than as points on the sphere.