Sec. 6.7 Coordinate treatment of harmonic ranges 95
6.6.3 Inequalities for closed half-planes...................
Let l≡ax+by+c= 0. Then the sets
{Z≡(x,y):ax+by+c≤ 0 }, (6.6.1)
{Z≡(x,y):ax+by+c≥ 0 }, (6.6.2)are the closed half-planes with common edge l.
Proof.LetZ 1 ≡(x 1 ,y 1 )be a point not inl,andletsl(Z 1 )=Z 2 ≡(x 2 ,y 2 ).Let
Z≡(x,y).Thenasin6.3.1,Z∈lif and only if|Z,Z 1 |^2 =|Z,Z 2 |^2 , and this occurs
when(x−x 1 )^2 +(y−y 1 )^2 =(x−x 2 )^2 +(y−y 2 )^2 , which simplifies to
2 (x 2 −x 1 )x+ 2 (y 2 −y 1 )y+x^21 +y^21 −x^22 −y^22 = 0.This is an equation forlandsoby6.3.1thereissomej=0 such that
ax+by+c=j[
2 (x 2 −x 1 )x+ 2 (y 2 −y 1 )y+x^21 +y^21 −x^22 −y^22]
By 4.3.4 the sets
{
Z≡(x,y):2(x 2 −x 1 )x+ 2 (y 2 −y 1 )y+x^21 +y^21 −x^22 −y^22 ≤ 0
}
, (6.6.3)
{
Z≡(x,y):2(x 2 −x 1 )x+ 2 (y 2 −y 1 )y+x^21 +y^21 −x^22 −y^22 ≥ 0}
, (6.6.4)
are the closed half-planes with edgel, as they correspond to|Z,Z 1 |≤|Z,Z 2 |and
|Z,Z 1 |≥|Z,Z 2 |, respectively. But when j>0, (6.6.1) and (6.6.3) coincide as do
(6.6.2) and (6.6.4), while whenj<0, (6.6.1) and (6.6.4) coincide as do (6.6.2) and
(6.6.3).
6.7 Coordinatetreatmentofharmonicranges...............
6.7.1 Newparametrisationofaline
As in 6.4.1, ifZ 1 ≡(x 1 ,y 1 ),Z 2 ≡(x 2 ,y 2 ),Z≡(x,y)wherex=x 1 +t(x 2 −x 1 ),y=
y 1 +t(y 2 −y 1 ),thenZ∈Z 1 Z 2 and
|Z 1 ,Z|^2 =[t(y 2 −y 1 )]^2 +[t(y 2 −y 1 )]^2 =t^2 |Z 1 ,Z 2 |^2 ,
|Z,Z 2 |^2 =[( 1 −t)(x 2 −x 1 )]^2 +[( 1 −t)(y 2 −y 1 )]^2 =( 1 −t)^2 |Z 1 ,Z 2 |^2 ,
|Z 1 ,Z|
|Z,Z 2 |=
∣∣
∣∣ t
1 −t∣∣
∣∣.
Accordingly, if we write 1 −tt=λwhereλ=0 and so havet= 1 +λλ,wehave
|Z 1 ,Z|
|Z,Z 2 |=|λ|.