1550251515-Classical_Complex_Analysis__Gonzalez_

Elementary Functions 245

3. d(z1,z2) = d(z2,z1).


4. d(z1,z3) ~ d(z1,z2) + d(z2,z3), where equality holds only if z1, z2, z 3
   are on the same line with z2 between z 1 and z 3.
   The direct justification of the strict triangle inequality is somewhat in-
   volved and will be postponed until a formula for the arc length in Poincare's
   model is derived.

The presence of the ideal points z;, z~ in the distance formula is rather

inconvenient, and we proceed to obtain an alternative formula involving

only the points z1 and Zz. It is clear that d(z1, z2) depends only on z1 and

Zz since z; and z~ are determined uniquely by z1 and Zz.

From (5.11-1) we have

az1 + b

W1 = CZ1 + d ' Wz = CZz +d

from which it follows that

and

z1 - z2 cz1 + d

=

z1 - Zz CZ1 + d

Hence

(5.11-5)

which shows that the expression J(z 1 -z 2 )/(z 1 - z 2 )J is invariant under the
group of transformations (5.11-1). In particular, for the points w 1 = i,

w 2 = ir, with r > 1 we get

so that

r=

Jz1 - z2l + Jz1 - z2l

Jz1 - z2l - lz1 - z2l

and in view of (5.11-3) and (5.11-4) we have

d(

• ) _ 1 Jz1 - z2 I + Jz1 - z2 J
  z 1 ,z 2 - n I
  lz1 - z2 - lz1 - z2 J
  which is the required formula.

(5.11-6)