1550251515-Classical_Complex_Analysis__Gonzalez_

64 Chapter^1

great circle of the Riemann sphere that contains those two points. This
arc is well determined except when the points Z 1 and Z 2 are the endpoints
of a diameter (i.e., when 1 + z 1 z 2 = O; see Exercises 1.7, problem 4), in
which case the spherical distance between them is % 7r (considering that
the Riemann sphere has radius! ). In the general case, let g be the great
circle passing through Z 1 and Z2, x(z1, z2) the length of the chord joining

Z1 and Z2, d 8 (z1, z2) the spherical distance, andμ the measure in radians

of the central angle Z 1 GZ 2 (Fig. 1.16). We have

and

so that

and

Hence

or

ds(z1,z2) =^1 / 2 μ

x^2 (zi, z2)

1-x^2 (z1,z2)

lz1 -z21^2

(1 + lz11^2 )(1 + lz21^2 ) - lz1 - z21^2

lz1 -z21^2

ll + .Z1z21^2

(1.17-16)

The spherical distance, as well as the chordal distance, are invariants
under certain groups of bilinear transformations discussed in Chapter 5.

Fig. 1.16