Sec. 7.6 Sensed distances 115
account. It can have negative as well as positive and zero values and it is related to
the distance asZ 1 Z (^2) ≤l=±|Z 1 ,Z 2 |or equivalently|Z 1 Z (^2) ≤l|=|Z 1 ,Z 2 |.
We note immediately the properties:
Z 1 Z 1 ≤l= 0 , (7.6.1)
Z 2 Z (^1) ≤l=−Z 1 Z (^2) ≤l, (7.6.2)
in all cases. We can add sensed distances on a line and have the striking property that
Z 1 Z (^2) ≤l+Z 2 Z (^3) ≤l=Z 1 Z (^3) ≤l, (7.6.3)
for allZ 1 ,Z 2 ,Z 3 ∈l.
This is easily seen to hold by (7.6.1) when any two of the three points coincide, as
e.g. whenZ 1 =Z 2 it amounts to 0+Z 1 Z 3 ≤l=Z 1 Z 3 ≤l. Suppose then thatZ 1 ,Z 2 ,Z 3
are all distinct and suppose first thatZ 1 ≤lZ 2. Then by 2.1.3 we have one of the cases
(a)Z 3 ≤lZ 1 ≤lZ 2 ,(b)Z 1 ≤lZ 3 ≤lZ 2 ,(c)Z 1 ≤lZ 2 ≤lZ 3.
In (a) we have
Z 1 Z 2 ≤l=|Z 1 ,Z 2 |,Z 2 Z 3 ≤l=−|Z 2 ,Z 3 |,Z 1 Z 3 ≤l=−|Z 1 ,Z 3 |,
and asZ 1 ∈[Z 3 ,Z 2 ],|Z 3 ,Z 1 |+|Z 1 ,Z 2 |=|Z 3 ,Z 2 |,whichis
−Z 1 Z 3 ≤l+Z 1 Z 2 ≤l=−Z 2 Z 3 ≤l.
In (b) we have
Z 1 Z (^2) ≤l=|Z 1 ,Z 2 |,Z 2 Z (^3) ≤l=−|Z 2 ,Z 3 |,Z 1 Z (^3) ≤l=|Z 1 ,Z 3 |,
and asZ 3 ∈[Z 1 ,Z 2 ],|Z 1 ,Z 3 |+|Z 3 ,Z 2 |=|Z 1 ,Z 2 |,whichisZ 1 Z (^3) ≤l−Z 2 Z (^3) ≤l=Z 1 Z 2 ≤l.
In (c) we have
Z 1 Z 2 ≤l=|Z 1 ,Z 2 |,Z 2 Z 3 ≤l=|Z 2 ,Z 3 |,Z 1 Z 3 ≤l=|Z 1 ,Z 3 |,
and asZ 2 ∈[Z 1 ,Z 3 ],|Z 1 ,Z 2 |+|Z 2 ,Z 3 |=|Z 1 ,Z 3 |,whichisZ 1 Z 2 ≤l+Z 2 Z 3 ≤l=Z 1 Z 3 ≤l.
Next suppose thatZ 2 ≤lZ 1. Then on interchangingZ 1 andZ 2 in the cases just
proved we haveZ 2 Z (^1) ≤l+Z 1 Z (^3) ≤l=Z 2 Z (^3) ≤l,forallZ 3 ∈land by (7.6.2) this gives
−Z 1 Z 2 ≤l+Z 1 Z 3 ≤l=Z 2 Z 3 ≤l. This completes the proof of (7.6.3) which shows that
addition of sensed distances on a line is much simpler than addition of distances.
We next relate sensed distances to the parametric equations oflnoted in
6.4.1,Corollary. Suppose thatW 0 ≡(u 0 ,v 0 ),W 1 ≡(u 1 ,v 1 )are distinct points onland
thatW 0 ≤lW 1. Then for pointsZ 1 ≡(x 1 ,y 1 ),Z 2 ≡(x 2 ,y 2 )onlwe have
x 1 = u 0 +s 1 (u 1 −u 0 ),y 1 =v 0 +s 1 (v 1 −v 0 ),
x 2 = u 0 +s 2 (u 1 −u 0 ),y 2 =v 0 +s 2 (v 1 −v 0 ),
and we recall thatZ 1 ≤lZ 2 if and only ifs 1 ≤s 2. Moreover, by the distance formula
|Z 1 ,Z 2 |=|s 2 −s 1 ||W 0 ,W 1 |.