If you consider hyperbolic $n$-space $H^n$, modeled by the open unit ball $B^n \subset \mathbb{R}^n$, then given any two distinct points $x_1$, $x_2$ in $H^n$, there is a natural way of identifying the unit tangent spheres $S_{x_1}$ and $S_{x_2}$ at $x_1$ and $x_2$ respectively. Start at $x_1$. Given a unit tangent vector $v$ at $x_1$, draw the geodesic ray starting at $x_1$ with initial velocity $v$, and define $f_1(v)$ to be the ideal point which is the limiting point of that geodesic ray. Then $f_1: S_{x_1} \to S_\infty$ is a diffeomorphism from $S_{x_1}$ onto the sphere at infinity.

Similarly, one may define the diffeomorphism $f_2: S_{x_2} \to S_\infty$. Then the composition $f_2^{-1} \circ f_1$ is a naturally defined diffeomorphism from $S_{x_1}$ onto $S_{x_2}$.

This is an example where we identify each "sphere of vision" (such as $S_{x_1}$, $S_{x_2}$) with the sphere at infinity.

Another example is Euclidean space $\mathbb{E}^n$. Assume that we compactify $\mathbb{E}^n$ by adding a point at infinity to each oriented direction (thus we add an $n-1$ dimensional sphere at infinity).

In this case, one may trivially identify each sphere of vision $S_x$, for $x \in \mathbb{E}^n$ with the sphere at infinity.

So here are two examples where one may identify each sphere of vision with the sphere at infinity.

Have similar geometric structures been studied before? Does this notion have a name please?

Edit (in reply to @RyanBudney): what I have in mind is something like this. Let $M$ be an $n$-dimensional manifold. Given $p \in M$, define

$$S_p = (T_p(M) \setminus \{0\})/\mathbb{R}_+$$

where $\mathbb{R}_+$ acts on $T_p(M)$ by scaling. Assume that you have a trivialization of the sphere bundle consisting of the unions of $S_p$, for $p \in M$. Then given any $v \in S^{n-1}$, one can associate to it a non-vanishing vector field on $M$, which is defined up to multiplication by a smooth positive function $f: M \to \mathbb{R}_+$. Thus this defines a foliation of $M$ by (unparametrized) curves.

So far, I have only used the trivialization of the bundle of $S_p$'s. But then I would like to make an additional assumption on $M$, namely that it has a compactification obtained by only adding an $S^{n-1}$ at infinity, such that given $v \in S^{n-1}$, the corresponding curves obtained by integrating the corresponding vector field, all go towards the same point on the sphere at infinity, say $f(v)$, and that the resulting map $f: S^{n-1} \to S^{n-1}_\infty$ is itself a diffeomorphism.

I am guessing one may define a point at infinity as an equivalence class of curves. I have seen this done before, but I don't remember in which article.