1550251515-Classical_Complex_Analysis__Gonzalez_

Elementary Functions 247

Next, letting dy = 0, then dx = 0, in (5.11-7) we get dsx = dx/y and

dsy = dy / y. This leads to the formula

dxdy

dA = dsxdsy = - 2 -

y

(5.11-10)

for the element of area in Cartesian coordinates in Poincare's model. As an
application, we shall find the area of a non-Euclidean triangle in hyperbolic
geometry.
The part of a non-Euclidean line through the points z 1 and z 2 , lying
between z 1 and z 2 , is called the non-Euclidean segment with endpoints z 1
and zz. By a non-Euclidean triangle with vertices z 1 , z 2 , z 3 is understood
the set of points bounded by the non-Euclidean segments determined by
each pair of vertices. The triangle is degenerated if all three vertices lie on
the same line, and the triangle is asymptotic if one or more vertices are
improper (ideal) points.
As a first step in evaluating the area of a triangle, we shall consider the
area of a triangle with two ideal vertices and a finite one. After preliminary
transformations (if necessary) the triangle will take the form of the shaded
strip S in Fig. 5.18, with two vertical sides and the third an arc of the
semicircle with center at the origin and radius one. Let M be the finite
vertex of the triangle, and call 8 the measure in radians of the interior
angle at M.
Then we have

fl

dxdy lcos 9 Joo dy 1cos 9 dx

A= - 2 - = dx 2 = ~ =7r-8
Y -1 v'1-x2 Y -1 - X
s
Thus the area of such a triangle is finite (in the hyperbolic metric) and
equals 7r - 8. As M --t +1, 8 --t 0 and we get that the area of a threefold

_____ ___.,.

x

Fig. 5.18