Graphics Reference
In-Depth Information
7.7 Rotating a Point about an Arbitrary Axis
Let us now consider two ways of developing a matrix for rotating a point
about an arbitrary axis. The first approach employs vector analysis and is
quite succinct. The second technique, however, is less analytical and relies on
matrices and trigonometric evaluation and is rather laborious. Fortunately,
they both arrive at the same result!
Figure 7.24 shows three views of the geometry associated with the task
at hand. The left-hand image illustrates the overall scenario; the middle im-
age illustrates a side elevation; whilst the right-hand image illustrates a plan
elevation.
w
P'
n
n
r
P'
r
a
P
r
N
P
N
N
a
r
Q
P
p'
p
n
Q
n
p
q
q
O
O
Fig. 7.24.
Three views of the geometry associated with rotating a point about an
arbitrary axis.
The axis of rotation is given by the unit vector
v
=
a
i
+
b
j
+
c
k
P
(
x
p
,y
p
,z
p
) is the point to be rotated by angle
α
and
P
(
x
p
,y
p
,z
p
)isthe
rotated point.
O
is the origin, whilst
p
and
p
are position vectors for
P
and
P
respec-
tively.
From Figure 7.24
p
=
−−→
ON
+
−−
NQ
+
−−→
QP
To find
−−→
ON
.
n
=
p
cos
θ
=
n
p
therefore
−−→
ON
=
n
=
n
(
n
.
p
)
