Graphics Reference
In-Depth Information
=[
e
2
+
e
3
]
Substituting
a
we obtain
⎡
⎣
⎡
⎣
⎤
⎦
⎤
⎦
=
⎡
⎣
⎤
⎦
2
3
2
3
1
3
1
0
1
0
1
1
1
3
2
3
2
3
−
−
2
3
1
3
2
3
−
which is correct.
You may also like to verify that the determinant of the matrix is 1.
12.6 Summary
It is very interesting to see the close relationship between quaternions and geomet-
ric algebra. It demonstrates that although it is possible to describe the low-level
arithmetic that actually does the work behind the scenes, such as a matrix, it is also
possible to invent objects such as quaternions or bivectors, trivectors, etc., that pro-
vide a conceptual high-level framework that allow mathematicians to work more
productively and creatively. In the end, Hamilton, Grassman and Clifford have pro-
vided us with some extraordinary mathematical inventions that have found their way
into computer graphics, and I hope that this chapter has shown you another way of
handling rotations.
12.6.1 Summary of Bivector Transforms
Reflecting a vector in a plane
v
=−ˆ
nv
n
.
ˆ
Rotating a vector using rotors
R
θ
vR
θ
v
=
where
sin
(θ/
2
)
B
R
θ
=
cos
(θ/
2
)
−
R
θ
=
sin
(θ/
2
)
B
.
cos
(θ/
2
)
+
Rotor as a matrix
⎡
⎤
⎡
⎤
2
(y
2
z
2
)
−
+
−
+
1
2
(xy
sz)
2
(xz
sy)
v
1
v
2
v
3
R
θ
vR
θ
=
⎣
2
(x
2
z
2
)
⎦
⎣
⎦
2
(xy
+
sz)
1
−
+
2
(yz
−
sx)
2
(x
2
+
y
2
)
2
(xz
−
sy)
2
(yz
+
sx)
1
−
⎡
⎤
⎡
⎤
2
(y
2
z
2
)
1
−
+
2
(xy
+
sz)
2
(xz
−
sy)
v
1
v
2
v
3
R
†
⎣
⎦
⎣
⎦
2
(x
2
z
2
)
θ
vR
θ
=
2
(xy
−
sz)
1
−
+
2
(yz
+
sx)
2
(x
2
+
y
2
)
2
(xz
+
sy)
2
(yz
−
sx)
1
−
