116 Circles; their basic properties Ch. 7
From these we conclude that
Z 1 Z (^2) ≤l=(s 2 −s 1 )|W 0 ,W 1 |. (7.6.4)
In particular the simplest case of parametric representation in relation to sensed dis-
tances is when we additionally take|W 0 ,W 1 |=1aswethenhaveZ 1 Z 2 ≤l=s 2 −s 1.
When we consider the reciprocal natural order onlwe note that
Z 1 Z (^2) ≥l=−Z 1 Z (^2) ≤l,
so that changing to the reciprocal natural order multiplies the value by−1. As well
as adding sensed distances on one line we can multiply or divide them. Now for
Z 1 ,Z 2 ,Z 3 ,Z 4 inl,
Z 3 Z (^4) ≥lZ 1 Z (^2) ≥l=−Z 3 Z (^4) ≤l(− 1 )Z 1 Z (^2) ≤l=Z 3 Z (^4) ≤lZ 1 Z (^2) ≤l,
so thissensed productis independent of which natural order is taken. Similarly,
whenZ 1 =Z 2 , we can take a ratio of sensed distances
Z 3 Z 4 ≤l
Z 1 Z 2 ≤l
=
−Z 3 Z 4 ≥l
−Z 1 Z 2 ≥l=
Z 3 Z 4 ≥l
Z 1 Z 2 ≥l=
s 4 −s 3
s 2 −s 1,
and see that thissensed ratiois independent of whichever of≤l,≥lis used. When
the linelis understood, we can relax our notation toZ 3 Z 4 Z 1 Z 2 andZZ^31 ZZ^42 for these
products and ratios.
If forZ 1 ,Z 2 ,Z∈lwe take the parametric equations
x=x 1 +t(x 2 −x 1 ),y=y 1 +t(y 2 −y 1 ),(t∈R),then we have that
x=u 0 +[s 1 +t(s 2 −s 1 )](u 1 −u 0 ),y=v 0 +[s 1 +t(s 2 −s 1 )](v 1 −v 0 ),and by (7.6.4) we have that
Z 1 Z≤l=t(s 2 −s 1 )|W 0 ,W 1 |,ZZ 2 ≤l=( 1 −t)(s 2 −s 1 )|W 0 ,W 1 |,and so
Z 1 Z
ZZ 2
=
t
1 −t. (7.6.5)
Our main utilisation of these concepts is through sensed ratios; for example
(Z 1 ,Z 2 ,Z 3 ,Z 4 )is a harmonic range whenZ 1 Z 3 /Z 3 Z 2 =−Z 1 Z 4 /Z 4 Z 2. It is convenient
to defer the details until Chapter 11. However we make one use of sensed products in
the next subsection.