1550251515-Classical_Complex_Analysis__Gonzalez_

248

.,,,.....-----..................

"' \

\ /

\/

-~----------~r------~-

Fig. 5.19

Chapter 5

asymptotic triangle (i.e., a triangle with all three vertices at ideal points)
is always rr.

Now consider the area of a triangle ABC with all three vertices at finite

points. Let A be the ideal point of the half-ray BC, B the ideal point of
the half-ray CA, and C* that of the half-ray AB(Fig. 5.19). Also, let a, /3,
'Y be the interior angles of triangle ABC, and S its area.
Since

ABC +ABC +A BC+ A BC =A BC*

and area ABC = rr-(rr-a)= a, and similarly for the other two doubly

asymptotic triangles, while A* B*C* is threefold asymptotic, we get

or

As a consequence, we have

a+f3+1<rr

showing that in Lobachevsky's geometry the sum of the angles of a triangle

is always less than rr, except when the triangle is threefold asymptotic, in

which case the area equals 1r.

Exercises 5.1

1. Assuming thew-plane superimposed on the z-plane, show geometrically
   the mapping determined by each of the following functions.
   (a) w = z - 2i (b) w = -iz
   (c) w=(l+i)z+3 (d) w=(l-i)z+(2+i)