Geometry with Trigonometry

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 ).