# SNCF metric

The following two examples of a metric space (one of which is a real
tree) obtained their name from the 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

$${d}_{P}(x,y):=\{\begin{array}{cc}0\hfill & \text{if}x=y\hfill \\ d(x,P)+d(P,y)\hfill & \text{otherwise.}\hfill \end{array}$$ |

It is easy to see that ${d}_{P}$ is a metric.

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 ${\mathbb{R}}^{n}$ with Euclidean norm
$\parallel \cdot {\parallel}_{2}$. Then the *SNCF metric* ${d}_{P}$ is defined by

$${d}_{P}(x,y):=\{\begin{array}{cc}{\parallel x-y\parallel}_{2}\hfill & \text{if}x\text{and}y\text{lie on the same ray from the origin}\hfill \\ {\parallel x\parallel}_{2}+{\parallel y\parallel}_{2}\hfill & \text{otherwise}\hfill \end{array}.$$ |

The metric space $({\mathbb{R}}^{n},{d}_{P})$ is, in addition, a real tree since
if $x$ and $y$ do not lie on the same http://planetmath.org/node/6962ray from $P$, the only arc in
$({\mathbb{R}}^{n},{d}_{P})$ joining $x$ and $y$ consists of the two ray http://planetmath.org/node/5783segments
$xP$ and $yP$. Other injections which are arcs in Euclidean^{}
${\mathbb{R}}^{n}$ do not remain continuous^{} in $({\mathbb{R}}^{n},{d}_{P})$.

Title | SNCF metric |
---|---|

Canonical name | SNCFMetric |

Date of creation | 2013-03-22 15:17:25 |

Last modified on | 2013-03-22 15:17:25 |

Owner | GrafZahl (9234) |

Last modified by | GrafZahl (9234) |

Numerical id | 5 |

Author | GrafZahl (9234) |

Entry type | Example |

Classification | msc 51M05 |

Classification | msc 97A20 |

Classification | msc 54E35 |

Related topic | RealTree |