French railway metric

Main lines of the railway centered on Paris in 1856

In mathematics , the French railroad metric is an unusual example of a metric .

Let there be a set of points in the plane and one fixed point. ${\displaystyle X}$ ${\displaystyle P}$

Then the French railway metric is defined by the function on ${\displaystyle X}$

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

${\displaystyle d(A,B)={\begin{cases}\|A-B\|&{\text{falls }}A,B{\text{ auf einer Geraden durch }}P{\text{ liegen, }}\\\|A-P\|+\|P-B\|&{\text{sonst}}.\end{cases}}}$

The construction can be generalized to any Euclidean or unitary vector spaces without difficulty .

The name is derived from the very centralized railway network in France, especially in the past, with almost all railway connections leading to Paris . The consequence of this was that z. B. had to accept a 400 km long detour via Paris when traveling by train from Strasbourg to Lyon , as there was no direct connection.

A metric is the mathematical generalization of distance . If the number of French cities with a rail connection to Paris is ( ), then, in analogy to the above metric, the distance traveled from city to city can be very long if there is no direct connection but only a connection via , even if the cities are linear to each other are close. The name SNCF metric is also in use, after the French national railway company SNCF . ${\displaystyle X}$ ${\displaystyle P}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle P}$

Another metric motivated by special architecture is the Manhattan metric .

literature

• Sur les groupes hyperboliques d'après Mikhael Gromov. (French) 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. Birkhauser Boston, Inc., Boston, MA, 1990. ISBN 0-8176-3508-4

Web links

• Plongement des espaces métriques et applications (page 5–6)