Graphics Reference
In-Depth Information
In Fig. 7.1,
n
c
k
is the unit vector representing the axis of rotation,
Px
p
y
p
z
p
is the point to be rotated by angle ,
P
x
p
y
p
z
p
is the rotated point,
p
and
p
are position vectors for P and P
, respectively,
O is the origin.
ˆ
=
a
i
+
b
j
+
The objective is to derive a vector equation describing
p
in terms of
n
p
, and .
So let's start by defining
p
in terms of vectors associated with the construction lines shown
in Fig. 7.1(a):
ˆ
+
−
QP
p
=
−
ON
+
−
NQ
We now define each of these vectors in turn.
To find
−
ON, we start by taking the dot product of
n
and
p
:
ˆ
n
ˆ
·
p
= ˆ
n
p
cos
=
p
cos
=
ON
Therefore,
−
ON
=
n
ˆ
·
p
n
ˆ
=
n
(7.1)
We use the right-angle triangle NQP
to find
−
NQ:
NQ
NP
r
NQ
NP
−
NQ
=
=
r
=
cos
r
(7.2)
To eliminate
r
in Eq. (7.2), we construct an equation defining
r
in terms of known vectors:
r
=
p
−
n
(7.3)
Substituting Eq. (7.1) in Eq. (7.3), we obtain
r
=
p
−
n
ˆ
·
p
n
ˆ
and
−
NQ
=
p
−
n
ˆ
·
p
n
cos
ˆ
To find
−
QP
, we exploit the fact that
w
[Fig. 7.1(c)] is perpendicular to the plane containing
n
ˆ
and
p
, such that
n
ˆ
×
p
=
w
where
w
= ˆ
n
p
sin
=
p
sin
(7.4)
