Radial Metric

red point

distance

0.4

angle

π

4

green point

distance

0.6

angle

3π

4

The radial metric

ρ

is defined on

×

as

ρ(x,y)=|x-y|

if

x

and

y

lie on the same line through the origin, and

|x|+|y|

otherwise. It looks like a centralized train network, and has been called the French railroad metric.

A neighborhood of a point

x

of radius

r

,

{y|ρ(x,y)<r}

, is either a line segment through

x

on the ray from the origin through

x

or the union of a segment from the origin through

x

and a disk centered at the origin.