Geometry with Trigonometry

Sec. 10.4 Sensed angles 153

and soz′−z 0 =( ̄z−z ̄ 0 )cis 2α. Hence

x′−x 0 +ı(y′−y 0 )=[x−x 0 −ı(y−y 0 )](cos2α+ısin2α),

so that

x′−x 0 =cos2α.(x−x 0 )+sin2α.(y−y 0 ),

y′−y 0 =sin2α.(x−x 0 )−cos2α.(y−y 0 ).

We can express this in matrix form as stated.
We denoteslbysα;Z 0 as well.

10.4 Sensedangles .............................

10.4.1 .....................................

Definition.ForF′=tO,Z 0 (F),letI 0 =tO,Z 0 (I).ThenifZ 1 =Z 0 ,Z 2 =Z 0 ,welet
θ 1 =∠F′I 0 Z 0 Z 1 andθ 2 =∠F′I 0 Z 0 Z 2 .Wedefinethesensed-angleFZ 1 Z 0 Z 2 to be
θ 2 −θ 1.

O I

J H 1

H 2

H 4 H 3

Z 0 I 0

J 0

Z 1

Z 2

θ 1

θ 2

θ 2 −θ 1

Figure 10.4.

Sensed angles have the following properties.
Throughout Z 0 ∼Fz 0 ,Z 1 ∼Fz 1 ,Z 2 ∼Fz 2.

(i)If the points Z 1 and Z 2 are both distinct from Z 0 , andφis the sensed-angle

FZ 1 Z 0 Z 2 ,then

Z 0 Z 2 F

Z 0 Z 1 F

=

z 2 −z 0

z 1 −z 0

=

|Z 0 ,Z 2 |

|Z 0 ,Z 1 |

cisφ.

(ii)The sensed-angleFZ 1 Z 0 Z 2 is wedge or reflex according asℑzz^21 −−zz^00 is positive

or negative, and this occurs according as^12 [(y 2 −y 0 )(x 1 −x 0 )−(x 2 −x 0 )(y 1 −

y 0 )]is positive or negative.