Complex Numbers 63
The length Q' Q of the chord joining Q and Q' is the Euclidean distance
between the spherical images of z and z'. Following Caratheodory it is
called the chordal distance between z and z' and denoted x(z, z'). Hence
1 lz-z'I
x(z, z) = vlzl2+lyilz'l2+1
(1.17-14)Example The ordinary distance from i to 1 + i is 1(1 + i) - ii = 1, while
the chordal distance is x(l + i,i) = 1/y'6.
The chordal distance has the same properties of the ordinary distance
between two points, namely:
- x(z, z') 2:: 0
- x(z,z') = 0 iff z = z'
- x(z,z') = x(z',z)
- x(z, z') :::; x(z, z") + x(z", z')
The first three properties follow at once from (1.17-14). The fourth
property, called the triangle inequality, follows simply by considering that
the spherical images Q, Q', and Q" of any three given complex numbers
z, z', and z" form a triangle in space, so that Q' Q :::; Q" Q + Q' Q". In fact,
since three points on the sphere are never collinear, we have in this case
the strict inequality Q'Q < Q"Q+Q'Q", or x(z,z') < x(z,z")+x(z",z').
This can also be established directly from (1.17-14) by using the result of
Exercises 1.2, problem 17.
The chordal distance satisfies the inequality
x(z, z'):::; 1since it is equivalent to the ordinary distance between two points of a
sphere of diameter 1.
. In the derivation of (1.17-14) we have assumed that z -=fa oo, z' -=fa oo:
However, the definition of chordal distance can be applied directly to the
pair z, oo(z "=fa oo), and one obtains
- 1
x(z,oo) = NQ =. I
V lzl2+1
(1.17-15)The same result follows from (1.17-14) by finding limlz'l-++oo x(z, z'). By
introducing the spherical representation of complex numbers, and making
use of the chordal distance, the point at infinity becomes indistinguishable
from any other complex number..
Instead of the chordal distance, it is convenient at times to consider
the spherical distance d 8 (z 1 , z 2 ), defined to be the length of the smallest
of the two arcs determined by the images Z1 and Z2 of z1 and z2 on the