238 Chapter 5The symmetry principle may be applied to solve the problem of finding a
bilinear transformation that maps a circle c into a given circler, carrying
a point z 0 on C into a point w 0 on r and a point z 1 not on C into a point
W1 not on r. Jn fact, by the symmetry principle, the symmetric point
zi with respect to c must then by carried into the symmetric point wr
(with respect to r) and those three pairs of points suffice to determine the
transformation.
Exampne Find the bilinear transformation that maps lz - 21 = 3 into the
unit circle lwl = 1, leaving the point -1 fixed and carrying 1 into 0. Here
z 0 = w 0 = -1, z1 = 1 and w 1 = 0. The symmetric of z 1 with respect to
the first circle is zi = [9/(1 - 2)] + 2 = -7, and the symmetric of w 1 = 0
with respect to the second circle is wr = oo. Hence the transformation
is of the form
z-1
w=k--z+7
for some k. Using the fixed point, we find k = 3 so the required
transformation is
z-1
w=3--z+7
The other fixed point of this transformation is -3. Alternatively, we may
obtain the result from
(-1,0,oo,w) = (-1,1,-7,z)
Theorem 5.9 A bilinear transformation, other than the identity, is
equivalent to the product of an even number of symmetries.
Proof As already noted, a bilinear transformation is a composition of some
of the following transformations: (a) translation, (b) inversion and symme-
try (in the usual sense), and (c) similitude (i.e., homotecy and rotation).
Since inversion with respect to a circle is the same as symmetry with re-
spect to that circle, and symmetry with respect to a line may be thought as
a special case of symmetry with respect to a circle (the line being regarded
as a circle through the point oo), it will suffice to consider the elementary
transformations (a) and (c).
Now, a translation w = z + b is equivalent to the product of two symme-
tries with respect to two parallel lines p 1 , p 2 perpendicular to the vector b
and at a distance %lbl of each other (Fig. 5.11), z being midway between
the parallels.
On the other hand, a rotation w = ei^9 is equivalent to the product of
two symmetries with respect to two concurrent lines, p, q at the center of