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fewer arithmetic operations when making computations (see Table 2 in [Tayl79]). A
third reason has to do with the fact that round-off errors cause numerical instability
in matrices so that one needs to renormalize them periodically, which is not as much
of a problem with the quaternions.
We finish this section with two useful conversion formulas. Although quaternions
may have technical advantages, other representations such as Euler angles (defined
in Section 2.5.1 in [AgoM05]) may provide a more intuitive way to describe rotations.
20.3.8 Proposition.
If [a,b,t] is the Euler representation of a rotation R of
R
3
with
center the origin, then R(z) =
qzq
-1
, where
q
is the quaternion defined by
q
=+ + +
ra b c
i
j
k
and
t
ba
t
ba
r
=
cos
cos
cos
+
sin
sin
sin
222
222
t
ba
t
ba
a
=
cos
cos
sin
-
sin
sin
cos
222
222
t
b a
t
ba
b
=
cos
sin
cos
+
sin
cos
sin
22 2
2 22
t
ba
t
ba
c
=
sin
cos
cos
-
cos
sin
sin
.
222
222
Conversely, if a rotation R of
R
3
with center the origin is defined by means of a
quaternion
q
=+ + +
ra b c,
i
j
k
then the Euler representation [a, b, t] of R is defined by
m
m
m
m
23
33
12
11
tan
a
=
,
tan
b
= -
m
,
and
tan
t
=
,
13
where
2
2
mr
=+-
=+
=-
=+
=+-.
221
22
22
22
221
a
11
mab
rc
12
mac
rb
13
mbc
ra
23
2
2
mr
c
33
Proof.
See [Kuip99].
Kuiper ([Kuip99]) also shows that the quaternion representation is very useful in
dealing with products of rotations and computing the rotation axis of the result.
