Sec. 10.5 Sensed-area 157
Note that
δF(Z 1 ,Z 2 ,Z 3 )=δF(Z 2 ,Z 3 ,Z 1 )=δF(Z 3 ,Z 1 ,Z 2 )
=−δF(Z 1 ,Z 3 ,Z 2 )=−δF(Z 2 ,Z 1 ,Z 3 )=−δF(Z 3 ,Z 2 ,Z 1 ),
so that its value is unchanged ifZ 1 ,Z 2 ,Z 3 are permuted cyclically, and its value is
multiplied by−1 if the order of these is changed.
We note also that 10.4.1(ii) can be restated as that the sensed-angleFZ 1 Z 0 Z 2 is
wedge or reflex according as
ℑ
z 2 −z 0
z 1 −z 0
=ℑ
Z 0 Z 2 F
Z 0 Z (^1) F
is positive or negative, and this occurs according asδF(Z 0 ,Z 1 ,Z 2 )is positive or neg-
ative.
10.5.3Abasicfeatureofsensed-area.....................
A basic feature of sensed-area is given by the following. Let the pointsZ 3 ≡(x 3 ,y 3 ),
Z 4 ≡(x 4 ,y 4 ),Z 5 ≡(x 5 ,y 5 )be such that
x 3 =( 1 −s)x 4 +sx 5 ,y 3 =( 1 −s)y 4 +sy 5 ,
for somes∈R. Then for allZ 1 ,Z 2 ,
δF(Z 1 ,Z 2 ,Z 3 )=( 1 −s)δF(Z 1 ,Z 2 ,Z 4 )+sδF(Z 1 ,Z 2 ,Z 5 ).
For
δF(Z 1 ,Z 2 ,Z 3 )=
1
2
det
⎛
⎝
x 1 y 1 1
x 2 y 2 1
( 1 −s)x 4 +sx 5 ( 1 −s)y 4 +sy 5 ( 1 −s)+s
⎞
⎠
=
1
2
det
⎛
⎝
x 1 y 1 1
x 2 y 2 1
( 1 −s)x 4 ( 1 −s)y 4 1 −s
⎞
⎠+^1
2
det
⎛
⎝
x 1 y 1 1
x 2 y 2 1
sx 5 sy 5 s
⎞
⎠
=
1
2
( 1 −s)det
⎛
⎝
x 1 y 1 1
x 2 y 2 1
x 4 y 4 1
⎞
⎠+^1
2
sdet
⎛
⎝
x 1 y 1 1
x 2 y 2 1
x 5 y 5 1
⎞
⎠
=( 1 −s)δF(Z 1 ,Z 2 ,Z 4 )+sδF(Z 1 ,Z 2 ,Z 5 ).
10.5.4 An identity for sensed-area .....................
An identity that we have for sensed-area is that for any pointsZ 1 ,Z 2 ,Z 3 ,Z 4 ,
δF(Z 4 ,Z 2 ,Z 3 )+δF(Z 4 ,Z 3 ,Z 1 )+δF(Z 4 ,Z 1 ,Z 2 )=δF(Z 1 ,Z 2 ,Z 3 ).