Geometry with Trigonometry

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

10.2 Complex-valueddistance .......................

10.2.1Complex-valueddistance .......................

The material in 7.6 is long established; we can generalise those concepts of sensed
distances and sensed ratios as follows.

Definition.LetF be a frame of reference forΠ.IfZ 1 ∼Fz 1 ,Z 2 ∼Fz 2 we

defineZ 1 Z (^2) F=z 2 −z 1 , and call this acomplex-valued distancefromZ 1 toZ 2 .We
then consider alsoZZ 13 ZZ^42 FF, a ratio of complex-valued distances orcomplex ratiowhen
Z 1 =Z 2.
We show that this latter reduces to the sensed ratio
Z 3 Z (^4) ≤l
Z 1 Z (^2) ≤l
whenZ 1 ,Z 2 ,Z 3 ,Z 4 are points of a linel. As in 7.6.1 we suppose thatlis the lineW 0 W 1
whereW 0 ≡(u 0 ,v 0 )andW 1 ≡(u 1 ,v 1 ), and has parametric equations
x=u 0 +s(u 1 −u 0 ),y=v 0 +s(v 1 −v 0 ).
By 10.1.1(ii)lthen has complex parametric equationz=w 0 +s(w 1 −w 0 ).If
Z 1 ,Z 2 ,Z 3 ,Z 4 have parameterss 1 ,s 2 ,s 3 ,s 4 , respectively, then
z 2 −z 1 =(z 2 −w 0 )−(z 1 −w 0 )=(s 2 −s 1 )(w 1 −w 0 ),z 4 −z 3 =(s 4 −s 3 )(w 1 −w 0 ),
and so
Z 3 Z 4 F
Z 1 Z (^2) F

=

s 4 −s 3

s 2 −s 1

By 7.6.1 this is equal to the sensed ratio. This shows that for four collinear points a
ratio of complex-valued distances reduces to the corresponding ratio of sensed dis-
tances.

COMMENT. We could make considerable use of this concept in our notation for
the remainder of this chapter but in fact we use it sparingly.

10.2.2 A complex-valued trigonometric function...............

ForZ 0 ∼Fz 0 andF′=tO,Z 0 (F),letI 0 =tO,Z 0 (I); we recall from 8.3 thatZ∼F′
z−z 0 .ThenifZ=Z 0 ,Z∼Fzandθ=∠F′I 0 Z 0 Z, by 9.2.2 we have

x−x 0 =rcosθ,y−y 0 =rsinθ,

wherer=|Z 0 ,Z|=|z−z 0 |. It follows thatz−z 0 =r(cosθ+ısinθ).