For points $a,b\in \mathbb{R}^n\setminus \{0\}$ denote $$d(a,b)=\frac{\|a-b\|}{\|a\|+\|b\|}.$$
[This](https://mathoverflow.net/questions/229907/inequality-from-a-point-in-plane-to-a-triangle-or-inequality-on-a-quadrilateral) question by Ritesh Ahuja (positive answered by Iosif Pinelis) says that $d$ is a metric. My questions are:


1) was it studied, does it have a name?


2) may it be viewed as a mix of Eucledian metrics in the sense 
$$
d(a,b)=\int \|F(a,t)-F(b,t)\|d\mu(t)
$$
for some function $F(a,t)$ on $a\in \mathbb{R^n}\setminus \{0\}$, $t\in (\Omega,\mu)$ is some measure space? For example, metric
$$
\frac{\|a-b\|}{\|a\|\cdot \|b\|},
$$
corresponding to Ptolemy inequality, is nothing but $\|F(a)-F(b)\|$, where $F(a)=a/\|a\|^2$ is inversion. Here $\Omega$ is a trivial 1-point measure space.



What is this distance about?
The first distance is a special case of the so-called [p-q relative metric](https://books.google.com/books?id=q_7FBAAAQBAJ&lpg=PA362&ots=QY9TbHWREw&dq=p%20q%20relative%20metric&pg=PA362#v=onepage&q=p%20q%20relative%20metric&f=false), which is in general defined on any Ptolemaic space as
\begin{equation\*}
 \frac{\|x-y\|}{\left(\tfrac12(\|x\|^p + \|y\|^p)\right)^{q/p}}.
\end{equation\*}
The book linked above provides more pointers (special cases include names like *p-relative metric* (Klamkin-Meir metric), *Schattschneider metric*, etc.)


