TITLE: What is this distance about?
QUESTION [5 upvotes]: For points $a,b\in \mathbb{R}^n\setminus \{0\}$ denote $$d(a,b)=\frac{\|a-b\|}{\|a\|+\|b\|}.$$
This 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.

REPLY [5 votes]: Later generalizations notwithstanding (as noted already), this particular one...
$$
d(a, b) = \frac{\lVert a - b \rVert}{\lVert a \rVert + \lVert b \rVert}
$$
...was originally called "a multiplicative metric" in the namesake article published in 1976 by Schattschneider, Doris J.. "A Multiplicative Metric". Mathematics Magazine 49.4 (1976): 203–205 where she proved that $d(a, b)$ is in fact a metric on $\mathbb R$.