Graphics Reference
In-Depth Information
Y
V
2
V
n
(1
−
t
)
q
b
(1
−
t
)
q
V
1
m
q
t
q
q
a
X
Fig. 8.12.
Vector
V
is derived from a parts of
V
1
and b parts of
V
2
.
of
V
2
:
V
=
a
V
1
+
b
V
2
(8.28)
Let's define the values of a and
b
such that they are a function of the separating
angle
θ
. Vector
V
is
tθ
from
V
1
and (1
−
t
)
θ
from
V
2
, and it is evident from
Figure 8.12 that using the sine rule
a
sin((1
b
sin(
tθ
)
t
)
θ
)
=
(8.29)
−
and furthermore
m
=
a
cos(
tθ
)
(8.30)
n
=
b
cos( (1
−
t
)
θ
)
(8.31)
m
+
n
= 1
(8.32)
From (8.29),
a
sin(
tθ
)
sin((1
− t
)
θ
)
b
=
(8.33)
and from (8.32) we get
a
cos(
tθ
)+
a
sin(
tθ
) cos((1
−
t
)
θ
)
=1
sin((1
−
t
)
θ
)
Solving for
a
we find
a
=
sin((1
t
)
θ
)
sin(
θ
)
−
and
b
=
sin(
tθ
)
sin(
θ
)
