Geometry with Trigonometry

160 Complex coordinates; sensed angles; angles between lines Ch. 10

10.7 Orientation of a triple of non-collinear points .....

10.7.1 .....................................

DefinitionWe say that an ordered triple(Z 0 ,Z 1 ,Z 2 )of non-collinear points ispos-
itively or negatively orientedwith respect toF according as the sensed angle
FZ 1 Z 0 Z 2 is wedge or reflex. By 10.4.1(ii) this occurs according asδF(Z 0 ,Z 1 ,Z 2 )
is positive or negative.

Definition.LetF=([O,I,[O,J)andF 1 =([Z 0 ,Z 1 ,[Z 0 ,Z 2 )be frames of
reference. We say thatF 1 ispositively or negatively orientedwith respect toF
according as(Z 0 ,Z 1 ,Z 2 )is positively or negatively oriented with respect toF.

The special isometries have the following effects on orientation:-

(i)Each translation preserves the orientations with respect toFof all

non-collinear triples.

(ii) Each rotation preserves the orientations with respect toFof all non-collinear

triples.

(iii)Each axial symmetry reverses the orientations with respect toFof all non-

collinear triples.

Proof.

(i) Letf=tZ 0 ,Z 1 andZ 2 ∼Fz 2 ,Z 3 ∼Fz 3 ,Z 4 ∼Fz 4 .Then

z′ 3 =z 3 +(z 1 −z 0 ), z′ 2 =z 2 +(z 1 −z 0 ),

so thatz′ 3 −z′ 2 =z 3 −z 2 , and similarlyz′ 4 −z′ 2 =z 4 −z 2 .Hence

z′ 4 −z′ 2

z′ 3 −z′ 2

=

z 4 −z 2

z 3 −z 2

,

and so by 10.4.1(ii) the result follows.
(ii) Letf=rα;Z 0. Then by 10.3.1

z′ 2 −z 0 =(z 2 −z 0 )cisα,z′ 3 −z 0 =(z 3 −z 0 )cisα,

and so
z′ 3 −z′ 2 =(z 3 −z 2 )cisα,z′ 4 −z′ 2 =(z 4 −z 2 )cisα.

Hence
z′ 4 −z′ 2
z′ 3 −z′ 2

=

z 4 −z 2

z 3 −z 2

,

and so by 10.4.1(ii) the result follows.
(iii) Letf=sα;Z 0. Then by 10.3.2

z′ 2 −z 0 =(z ̄ 2 −z ̄ 0 )cis 2α,z′ 3 −z 0 =(z ̄ 3 −z ̄ 0 )cis 2α,