Graphics Programs Reference
In-Depth Information
I
,
J
,and
K
are three different solutions of the matrix equation
X
2
=
−
U
and should
be considered the square roots of minus the identity matrix.
Quaternions can also be viewed as elements of a four-dimensional
vector space
,one
of whose bases are given by
⎛
⎞
⎛
⎞
0100
−
00 0
−
1
1000
0001
00
00
10
01 0 0
10 0 0
−
⎝
⎠
⎝
⎠
i
=
,
j
=
,
−
10
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
00
10
00 01
10 00
0
−
1000
0100
0010
0001
k
=
,
1
=
.
−
100
Quaternions satisfy the following identities, also known as Hamilton's Rules,
i
2
=
j
2
=
k
2
=
−
1
,
ij
=
−
ji
=
k
,
jk
=
−
kj
=
i
,
ki
=
−
ik
=
j
.
They have the following multiplication table:
1i
j
k
1
1i
j
k
i
i
−
1k
−
j
j
j
−
k
−
1i
k
kj
−
i
−
1
The eight quaternions
±
1
,
±
i
,
±
j
,and
±
k
form a group of order 8 with multipli-
cation as the group operation.
Quaternions can also be interpreted as a combination of a scalar and a vector. They
are consequently closely related to 4-vectors. Using this interpretation, a quaternion
q
can be represented as the sum
q
=
w
+
x
i
+
y
j
+
z
k
, the 4-tuples (
x, y, z, w
)and
(
w, x, y, z
), or the pair [
s,
v
], where
s
=
w
and
v
=(
x, y, z
).
Theconjugatequaternionisgivenby
q
∗
=
w
−
x
i
−
y
j
−
z
k
. The sum or difference
of two quaternions is the obvious
q
1
±
q
2
=(
w
1
+
w
2
)
±
(
x
1
+
x
2
)
i
±
(
y
1
+
y
2
)
j
±
(
z
1
+
z
2
)
k
=[
s
1
±
s
2
,
(
v
1
±
v
2
)]
,
and the product is the nonobvious
q
1
·
q
2
=(
w
1
w
2
−
x
1
x
2
−
y
1
y
2
−
z
1
z
2
)+(
w
1
x
2
+
x
1
w
2
+
y
1
z
2
−
z
1
y
2
)
i
+(
w
1
y
2
−
x
1
z
2
+
y
1
w
2
+
z
1
x
2
)
j
+(
w
1
z
2
+
x
1
y
2
−
y
1
x
2
+
z
1
w
2
)
k
=[(
s
1
s
2
−
v
1
•
v
2
)
,
(
s
1
v
2
+
s
2
v
1
+
v
1
×
v
2
)]
.
A quaternion product is associative (i.e., (
q
1
q
2
)
q
3
=
q
1
(
q
2
q
3
)) but not commutative.
Search WWH ::
Custom Search