Geometry with Trigonometry

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

10.5 Sensed-area ..............................

10.5.1 .....................................

For points Z 0 ≡F(x 0 ,y 0 ),Z 1 ≡F(x 1 ,y 1 ),Z 2 ≡F(x 2 ,y 2 )such that Z 1 =Z 0 ,Z 2 =Z 0 ,
andθ=FZ 1 Z 0 Z 2 we have

(i)

1

2 |Z^0 ,Z^1 ||Z^0 ,Z^2 |sinθ=

1

2 [(x^1 −x^0 )(y^2 −y^0 )−(x^2 −x^0 )(y^1 −y^0 )],

(ii)

1

2 |Z^0 ,Z^1 ||Z^0 ,Z^2 |cosθ=

1

2 [(x^1 −x^0 )(x^2 −x^0 )+(y^1 −y^0 )(y^2 −y^0 )].

Proof. By 8.3 and 9.2.2, ifk 1 =|Z 0 ,Z 1 |,k 2 =|Z 0 ,Z 2 |,then

x 1 −x 0 =k 1 cosθ 1 ,y 1 −y 0 =k 1 sinθ 1 ,

x 2 −x 0 =k 2 cosθ 2 ,y 2 −y 0 =k 2 sinθ 2.

Then by 9.3.3 and 9.3.4,

k 1 k 2 sin(θ 2 −θ 1 )=k 2 sinθ 2 k 1 cosθ 1 −k 2 cosθ 2 k 1 sinθ 1

=(y 2 −y 0 )(x 1 −x 0 )−(x 2 −x 0 )(y 1 −y 0 ).

Similarly

k 1 k 2 cos(θ 2 −θ 1 )=k 2 cosθ 2 k 1 cosθ 1 +k 2 sinθ 2 k 1 sinθ 1

=(x 2 −x 0 )(x 1 −x 0 )+(y 2 −y 0 )(y 1 −y 0 ).

10.5.2 Sensed-area of a triangle .....................

For an ordered triple of points(Z 1 ,Z 2 ,Z 3 )of points and a frame of referenceF,if
Z 1 ≡F(x 1 ,y 1 ),Z 2 ≡F(x 2 ,y 2 )andZ 3 ≡F(x 3 ,y 3 ), we recall from 6.6.2 and 10.5.1(i)
δF(Z 1 ,Z 2 ,Z 3 )defined by the formula

δF(Z 1 ,Z 2 ,Z 3 )=^12 [x 1 (y 2 −y 3 )−y 1 (x 2 −x 3 )+x 2 y 3 −x 3 y 2 ]

=^12 [(x 2 −x 1 )(y 3 −y 1 )−(x 3 −x 1 )(y 2 −y 1 )]

=

1

2

det

⎛

⎝

x 1 y 1 1

x 2 y 2 1

x 3 y 3 1

⎞

⎠.

By 6.6.2, whenZ 1 ,Z 2 ,Z 3 are non-collinear|δF(Z 1 ,Z 2 ,Z 3 )|is equal to the area of the
triangle[Z 1 ,Z 2 ,Z 3 ]. In this case we refer toδF(Z 1 ,Z 2 ,Z 3 )as thesensed-areaof the
triangle[Z 1 ,Z 2 ,Z 3 ], with the order of vertices(Z 1 ,Z 2 ,Z 3 ). This was first introduced
by Möbius in 1827.