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.