Graphics Reference
In-Depth Information
20.3
Quaternions as Transformations
The previous section discussed the basic properties of quaternions. Now we come
to the most important aspect of quaternions from the point of view of geometric
modeling, namely, their close relationship to rotations in 3-space.
20.3.1
Proposition.
If
q
is a nonreal unit quaternion, then
q
can be represented
uniquely in the form
q
=
cos
q
+
sin
q
n
,
(20.3)
where 0 £q£pand
n
is a unit quaternion with
n
2
=-1.
Proof.
Let
q
= r +
v
, r Œ
R
,
v
Œ
R
3
. Since
q
has unit norm, -1 £ r £ 1 and so there
is a unique q in the stated range with cos q=r. It follows from equation (20.2a) that
2
2
1
==+
q
r
v
,
or
v
2
2
2
=-
r
=
sin
q
But sin q≥0 for 0 £q£p, so that |
v
| = sin q. Since
q
is not real,
v
π
0
and sin qπ0.
Let
1
sin
n
=
v
.
q
Finally, equation (20.2b) implies that
n
2
=-1.
20.3.2
Corollary.
Any quaternion
q
can be represented uniquely in the form
(
)
q
=
r cos
q
+
sin
q
n
,
(20.4)
where 0 £q£pand
n
is a unit quaternion with
n
2
=-1.
Proof.
This is obvious because r is just |
q
|.
The representation (20.4) should remind us of the polar form representation of
complex numbers.
Definition.
The right hand side of equation (20.4) is called the
polar form
represen-
tation of the quaternion
q
.
20.3.3
Example.
To find the polar form of the quaternion
q
= 1 +
i
+
j
+
k
.
Solution.
Since |
q
| = 2 and cos q=1/2 implies that q=p/3, we get that
