1549055259-Ubiquitous_Quasidisk__The__Gehring_

3.10. QUADRILATERALS 47

i a+i

R

f

0 a

FIGURE 3.2

The modulus of a quadrilateral Q can also be given in terms of the modulus of
the family of curves joining opposite sides of Q.
LEMMA 3.10.1. If Q = D(z 1 , z 2 , z3, z 4 ) is a quadrilateral, then
(3.10.2) mod(Q) = mod(r)
where I' is the family of arcs in Q which join the sides of Q with endpoints z 1 , z 2
and z3, Z4.
PROOF. Since mod(Q) and mod(I') are conformally invariant, it suffices to con-
sider the case where Q is the rectangle R = R(O, a, a+ i , i). The desired conclusion
then follows from Lemma 1.3.1. 0

Recall that the cross ra tio of four points z 1, z 2 , z3, z4 in R
2
is denoted by
[z 1 , z 2 , z3, z4]. By choosing a = 1 in the rectangle R = R(O, a, a+ i , i) we im-
mediately have the following corollary to Lemma 3.10.l whenever D is a disk or a
half-plane.

COROLLARY 3.10.3. Let Q be the quadrilateral B(zi, z 2 , z3, z 4 ) or H(z 1 , z2, Z3, z 4 ).
Then we have that
mod(Q) = 1
if and only if
[z 1,z2,z3,z4] =2.
The next result yields a more general relation between the modulus of a quadri-
lateral and the cross ratio of its vertices in the disk or half-plane setting.

(3.10.5)

where

(3. 10 .6)

7r K ( vr=r:2)

μ(r) = 2 K(r)

with K ( r) the complete elliptic integral of the first kind,

fl dt

K(r) =Jo J(l - t^2 )(1 - r^2 t^2 )'