Graphics Reference
In-Depth Information
q
q
q
=
cos
+
sin
n
Œ
H
,
2
2
then R =s q .
Proposition 20.3.5 defines a map
3
()
r :
S O
q
Æ
3
,
(20.10)
Æ M
q
which is an important and well-known map in topology. This map is two-to-one
because r( q ) =r(- q ). It is onto by Proposition 20.3.6. Using Proposition 20.3.4(1) we
also see that the map is multiplicative because the product of two unit quaternions
gets mapped to the composition of their associated rotations of R 3 . This, by the way,
gives an indirect proof of the fact that the composition of two rotations about two
lines is a rotation about another line. Finally, the two propositions show us how one
can easily pass back and forth between the matrix representation of a rotation in R 3
that fixes the origin and its representation as a unit quaternion.
20.3.7 Example. To find the formula for the rotation R about the z-axis through
an angle of p/2 in terms of quaternions and to compute its action on i .
Solution.
We use the notation in Proposition 20.3.5. Now n = k and 2q=p/2.
Therefore,
1
2
1
2
q
=+
k
,
1
2
1
2
-
1
qq
==
-
k
(
using the fact that
q
=
1
and Proposition 20.2.4 3
()
)
,
and
1
2
() =
-1
(
)
(
)
R zqzq
=
1
+
kz k
1
-
.
It is easy to show that R( i ) = j , which reestablishes the fact that R sends the x-axis to
the y-axis. The matrix for R is
010
100
001
Ê
ˆ
Á
Á
˜
˜
M q =-
.
Ë
¯
The fact that elements of the special orthogonal group SO (3) can be represented
as quaternions is significant to computer graphics because quaternions make for
a better representation for certain applications (see, for example, [Tayl79] and
[YanF64]). One reason is that a quaternion takes less space, namely, one only needs
to store four real numbers versus nine for a matrix. Another reason is that one needs
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