1550251515-Classical_Complex_Analysis__Gonzalez_

240 Chapter^5

a bilinear transformation is equiva:lent to the product of an even number

of symmetries.

5.10 ORIENTATION OF A CffiCLE

An ordered triple of distinct points (zi, z 2 , z 3 ] on a circle C determines an

orientation on C. This orientation can be indicated geometrically by an

arrow pointing from z1 over z 2 to Z3 (Fig. 5.14). The ordered triple (z1, z3

z 2 ] then determines the opposite orientation.

Analytically, one orientation can be distinguished from the opposite by

considering the sign of the expression

1 1 1

which represents four times the signed area of the triangle with vertices z 1 ,

z 2 , Z3 (see Exercises 1.4, problem 7; also, compare with Section 1.4). The

+ sign corresponds to the usual positive (counterclockwise) orientation of

C, the - sign to its negative (clockwise) orientation.

Now consider a fourth point z and the bilinear transformation T that

carries z 1 into 1, z 2 into O, and z 3 into oo. Then we have, as seen in

Section 5.5,

(z, z1, z2, z 3 ) = (Tz, 1, 0, oo) = Tz (5.10-1)

Since any bilinear transformation carries circles into circles, the oriented

circle determined by (z 1 , z 2 z 3 ] goes over the oriented circle determined by

[1, O, oo], i.e., to the real axis with the orientation shown by the arrow in

Fig. 5.14. Hence the fourth point z goes over to the reals iff z lies on C.

In other terms, Tz is real, and so is the cross ratio (z,z 1 , z2 z 3 ) iff z EC.

The position of Tz on the real axis depends on the position of z on the

circle C. If z lies in the arc from Z3 to z 1 that does not contain z 2 (as in

Fig. 5.14), then Tz > 1; if z lies between z 1 and z2 (in the arc that does

Tz

c cc

Fig. 5.14