Graphics Reference
In-Depth Information
R
β
p
−
rotates
p
,
β
i.e. clockwise
pR
†
β
rotates
p
,β
i.e. anticlockwise
and
pR
β
R
β
p
=
R
β
p
.
Using the rotor
R
β
in a single-sided transformation only works for vectors in the
plane of rotation, which satisfies everything we do in 2D. However, in 3D we have
to employ a double-sided, half-angle formula of the form
R
β
pR
β
, which is covered
in Chap. 11.
pR
β
=
7.5.4 Rotate a Point About an Arbitrary Point
Earlier in this chapter we developed a transform for rotating a point about an ar-
bitrary point. Let's show how we can approach the same problem using geometric
algebra. Figure
7.8
shows the geometry describing how the point
P
is rotated
β
about
T
to
P
, and by inspection we can write
p
=
t
+
R
β
(
p
−
t
) .
Using the previous example, where
T
=
(
1
,
1
), P
=
(
2
,
1
)
and
β
=
90° we have
R
90
°
=
cos 90°
−
sin 90°
e
12
=−
e
12
t
=
e
1
+
e
2
p
=
2
e
1
+
e
2
p
=
e
1
+
e
2
−
e
12
(
2
e
1
+
e
2
−
e
1
−
e
2
)
=
e
1
+
e
2
−
e
12
e
1
=
e
1
+
e
2
+
e
2
=
e
1
+
2
e
2
which is correct.
Fig. 7.8
Using a rotor
R
β
to
rotate
P
about
T
