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 dP with respect to P is defined by

dP(x,y):={0if x=yd(x,P)+d(P,y)otherwise.

It is easy to see that dP 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 n with Euclidean norm 2. Then the SNCF metric dP is defined by

dP(x,y):={x-y2if x and y lie on the same ray from the originx2+y2otherwise.

The metric space (n,dP) is, in addition, a real tree since if x and y do not lie on the same from P, the only arc in (n,dP) joining x and y consists of the two ray xP and yP. Other injections which are arcs in EuclideanPlanetmathPlanetmath n do not remain continuousPlanetmathPlanetmath in (n,dP).

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