Graphics Reference
In-Depth Information
Fig. 8.10
The
X
Y
axial
system is rotated 90° and
translated
(
10
,
5
)
8.5.3 Rotated and Translated Frame of Reference
Earlier, we saw how two inverse transforms are used to compute the coordinates of a
point in a rotated and translated frame of reference. We can achieve the same result
using multivectors as follows.
We begin with point
P
and its frame of reference
XY
. The first step is to establish
a translated frame of reference
X
T
Y
T
with position vector
t
. Which means that
p
T
=
p
−
t
.
(8.3)
Next, as shown in Fig.
8.10
, we rotate
p
T
by
−
β
to effectively rotate the frame of
reference
X
T
Y
T
to
X
Y
. Which means that
p
=
R
β
p
T
.
(8.4)
Substituting (
8.3
)in(
8.4
)wehave
p
=
R
β
(
p
−
t
)
or
sin
β
e
12
)
(x
−
t
x
)
e
1
+
(y
−
t
y
)
e
2
.
p
=
(
cos
β
+
(8.5)
Let's test (
8.5
) using the same values in the previous example where
β
=
90°
(t
x
,t
y
)
=
(
10
,
5
)
(x, y)
(
9
,
6
)
p
=
cos 90°
=
sin 90°
e
12
(
9
5
)
e
2
+
−
10
)
e
1
+
(
6
−
=
e
12
(
−
e
1
+
e
2
)
=
e
1
+
e
2
which makes
P
=
(
1
,
1
)
the same as (
8.2
).
Although multivectors provide an alternative way of solving vector-based prob-
lems, they still have a matrix background. For example, expanding (
8.5
)wehave
