The following two examples of a metric space (one of which is a real tree) obtained their name from the operator of the French railway system. Especially malicious rumour has it that if you want to go by train from $x$ to $y$ in France, the most efficient solution is to reduce the problem to going from $x$ to Paris and then from Paris to $y$ .

Since their discovery, the intrinsic laws of the French way of going by train have made it around the world and reached the late-afternoon tutorials of first-term mathematics courses in an effort to lighten the moods in the guise of the following definition:

Definition 1 (SNCF metric)   Let $P$ be a point in a metric space $(F,d)$ . Then the SNCF metric $d_P$ with respect to $P$ is defined by

Now, what if the train from $x$ to Paris stops over in $y$ during the ride (or the other way round)? Sure, Paris is a beautiful city, but you wouldn't always want to go there and back again. To implement this, the geometric notion of ``$y$ lies on the straight line defined by $x$ and $P$ '' is required, so the definition becomes more specialised:

Definition 2 (SNCF metric, enhanced version)   Let $P$ be the origin in the space $\mbb{R}^n$ with Euclidean norm $\|\cdot\|_2$ . Then the SNCF metric $d_P$ is defined by