Graphics Reference
In-Depth Information
Next,
R
θ
v
2
e
2
R
θ
=
−
x
e
23
−
y
e
31
−
+
x
e
23
+
y
e
31
+
(s
z
e
12
)v
2
e
2
(s
z
e
12
)
=
v
2
(s
e
2
+
x
e
3
−
y
e
123
+
z
e
1
)(s
+
x
e
23
+
y
e
31
+
z
e
12
)
=
v
2
2
(xy
−
sz)
e
1
+
s
2
−
z
2
e
2
+
2
(yz
+
sx)
e
3
.
−
x
2
+
y
2
Substituting
s
2
y
2
x
2
z
2
+
=
1
−
−
we have
R
θ
v
2
e
2
R
†
v
2
2
(xy
sz)
e
1
+
1
2
x
2
z
2
e
2
+
sx)
e
3
.
θ
=
−
−
+
2
(yz
+
Next,
R
θ
v
3
e
3
R
θ
=
z
e
12
)
=
v
3
(s
e
3
−
x
e
2
+
y
e
1
−
z
e
123
)(s
+
x
e
23
+
y
e
31
+
z
e
12
)
=
(s
−
x
e
23
−
y
e
31
−
z
e
12
)v
3
e
3
(s
+
x
e
23
+
y
e
31
+
v
3
2
(xz
sx)
e
2
+
s
2
z
2
e
3
.
x
2
y
2
+
sy)
e
1
+
2
(yz
−
−
−
+
Substituting
s
2
z
2
x
2
y
2
+
=
1
−
−
we have
R
θ
v
3
e
3
R
θ
=
v
3
2
(xz
−
sy)
e
1
+
2
(yz
−
sx)
e
2
+
1
2
x
2
+
y
2
e
3
.
−
Therefore,
R
θ
vR
θ
=
R
v
1
e
1
R
†
R
v
2
e
2
R
†
R
v
3
e
3
R
†
+
+
or as a matrix
⎡
⎤
⎡
⎤
⎡
⎤
v
1
v
2
v
3
2
(y
2
+
z
2
)
1
−
2
(xy
−
sz)
2
(xz
+
sy)
v
1
v
2
v
3
⎣
⎦
=
⎣
⎦
⎣
⎦
2
(x
2
z
2
)
2
(xy
+
sz)
1
−
+
2
(yz
−
sx)
2
(x
2
y
2
)
2
(xz
−
sy)
2
(yz
+
sx)
1
−
+
which is the same matrix representing the quaternion triple
qpq
−
1
.
The reader should not be put off by the above algebraic proof. It has been in-
cluded to demonstrate that bivector rotors behave just like quaternions and are rep-
resented by identical matrices.
You may wish to investigate the matrix for the reverse rotor triple
R
θ
pR
θ
, which
you will discover is
⎡
⎣
⎤
⎦
=
⎡
⎤
⎡
⎤
v
1
v
2
v
3
2
(y
2
z
2
)
−
+
+
−
1
2
(xy
sz)
2
(xz
sy)
v
1
v
2
v
3
⎣
2
(x
2
z
2
)
⎦
⎣
⎦
2
(xy
−
sz)
1
−
+
2
(yz
+
sx)
2
(x
2
y
2
)
2
(xz
+
sy)
2
(yz
−
sx)
1
−
+
and is the transpose of the above matrix for
R
θ
vR
θ
. Thus the matrices confirm that
