French railway metrics

In mathmetics the French railway metric is a special example of a metric. It is also called SNCF metric, derived from the national French railway company SNCF.

Let $X$ be a set of points in the plane and $P$ a set point and $A$ , $B$ ∈ $X$ .

Then, the French railway metric is defined on $X$ through the function

d\colon X\times X\to \mathbb {R}

d(A,B)={\begin{cases}\|A-B\|&{\text{if }}A{\text{ and }}B{\text{ are on a line through }}P{\text{, }}\\\|A-P\|+\|P-B\|&{\text{in all other cases}}.\end{cases}}

This construction can easily be generalized in the case of Euclidean or Unitarian vector spaces or even in star domains.

The name originates from the French railway system, that was and still is mainly centralized on Paris. As a result, a passenger travelling from Strasbourg to Lyon used to have to take a detour of 400 km over Paris, since in contrast to nowadays there was no direct train route.

A metric is the mathematical generalization of distance. Let $X$ be the set of French cities with railway routes to Paris ( $P$ ). As in the general case above, the traveled distance from city $A$ to $B$ can be very long, due to the lack of a direct connection from $A$ to $B$ . The only existing one was the one going through $P$ , even if cities were close to each other.

The Manhattan metric is another metric derived from special urban planning.

Literature

Sur les groupes hyperboliques d'après Mikhael Gromov. Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. Edited by É. Ghys and P. de la Harpe. Progress in Mathematics, 83. Birkhäuser, Boston 1990, ISBN 0-8176-3508-4.

References

Plongement des espaces métriques et applications (pages 5–6)