152 Complex coordinates; sensed angles; angles between lines Ch. 10
Let l be the line Z 0 Z 1 ,Z 0 ∼Fz,F′=tO,Z 0 (F),I 0 =tO,Z 0 (I)andα=∠F′I 0 Z 0 Z 1.
Then sl(Z)=Z′where
Z∼Fz,Z′∼Fz′,z′−z 0 =(z ̄−z ̄ 0 )cis 2α,so that slhas the real coordinates form
x′−x 0 =cos2α.(x−x 0 )+sin2α.(y−y 0 ),
y′−y 0 =sin2α.(x−x 0 )−cos2α.(y−y 0 ),and so has the matrix form
(
x′−x 0
y′−y 0
)
=
(
cos2α sin2α
sin2α −cos2α)(
x−x 0
y−y 0)
.
Proof. To find a formula forZ′=sl(Z)we first show that ifW=πl(Z)andW∼F
wthen
w−z 0 =ℜ[
z−z 0
z 1 −z 0]
(z 1 −z 0 ),z−w=ıℑ[
z−z 0
z 1 −z 0(z 1 −z 0 )]
.
To start on this we note that
z−z 0 =z−z 0
z 1 −z 0(z 1 −z 0 )=[
ℜ
z−z 0
z 1 −z 0]
(z 1 −z 0 )+ı[
ℑ
z−z 0
z 1 −z 0]
(z 1 −z 0 ).If we now definewby
w−z 0 =[
ℜ
z−z 0
z 1 −z 0]
(z 1 −z 0 )thenW∈Z 0 Z 1 asw−z 0 is a real multiple ofz 1 −z 0 .Butthen
z−w=ı[
ℑ
z−z 0
z 1 −z 0]
(z 1 −z 0 ),soWis on a line throughZwhich is perpendicular toZ 0 Z 1. ThusWis the foot of the
perpendicular fromZtoZ 0 Z 1.
From this, asz′+z= 2 w,wehavez′−z=z′−w−(z−w)=− 2 (z−w)so
z′−z=− 2 ı[
ℑ
z−z 0
z 1 −z 0]
(z 1 −z 0 ).Asz 1 −z 0 =kcisαfor somek>0, we then have
z′−z=− 2 ı[
ℑz−z^0
kcisα]
kcisα=− 2 ı{ℑ[(z−z 0 )cis(−α)]}cisα=−[(z−z 0 )cis(−α)−(z ̄−z ̄ 0 )cisα]cisα=−(z−z 0 )+( ̄z−z ̄ 0 )cis 2α