1550251515-Classical_Complex_Analysis__Gonzalez_

230 Chapter^5

(a}

(b)

Fig. 5.6

Since a circle about the origin has equation IZI = r, the corresponding

circle C 2 has the equation

1

~1=r Z ~ Z2

so it is a circle of Apollonius with limit points z 1 and z 2 , i.e., it is the locus
of points whose ratio of distances from z1 and z2 is a constant. This circle
is transformed by L into another circle of the same family, namely,

I

w-z1 I= Rr

w-z2

The figure formed by all circles C1 and C2 is called the Steiner configuration

(or the circular net) determined by z 1 and z 2 (see Fig. 5.6b).

The bilinear transformation of the type just discussed are called hyper-

bolic (fol.lowing F. Klein). Since for these transformations M = R =f 1,