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?