232 Chapter 5
mapped into themselves. The points z 1 and z 2 are inverses of each
other with respect to any one of those orthogonal circle (Fig. 5.8).
The bilinear transformations of this type are called elliptic. For this
case (5.7-5) gives
or
(a+ d)2 = eilJ + e-ie + 2 = [e(1/2)i1J + e-(1/2)i1J]2
()
= 4cos^2 '2()a+d= ±2cos '2
Hence for elliptic transformations a + d is real, and
since 0 -=/:- 2k7r.
(5.8-4)Whenever 0 = 27rm/n, with gcd(m, n) = 1, we have n8 = 2m7r and
Mn = e^2 m,,.i = 1. Thus by applying the transformation n times, each
point returns to its original position, and the transformation is said to be
periodic with period n. Only elliptic transformations for which 0 /27r is
fractional have this property.
Example If 0 = 7r = %(27r), the transformation has period 2. In this
case M = e;,,. = -1 and a+ d = 0.
Fig. 5.8