It is noticeable that, while regular complex numbers of unit length z = e
iθ
can encode
rotations in the 2D plane (with one complex product, x
0
= z ·x), “extended complex
numbers” or quaternions of unit length q = e
(u
x
i+u
y
j+u
z
k)θ/2
encode rotations in the 3D
space (with a double quaternion product, x
0
= q ⊗ x ⊗ q
∗
, as we explain later in this
document).
CAUTION: Not all quaternion definitions are the same. Some authors write the
products as ib instead of bi, and therefore they get the property k = ji = −ij, which results
in ijk = 1 and a left-handed quaternion. Also, many authors place the real part at the end
position, yielding Q = ia+jb+kc+d. These choices have no fundamental implications but
make the whole formulation different in the details. Please refer to Section 3 for further
explanations and disambiguation.
CAUTION: There are additional conventions that also make the formulation different
in details. They concern the “meaning” or “interpretation” we give to the rotation op-
erators, either rotating vectors or rotating reference frames –which, essentially, constitute
opposite operations. Refer also to Section 3 for further explanations and disambiguation.
NOTE: Among the different conventions exposed above, this document concentrates on
the Hamilton convention, whose most remarkable property is the definition (2). A proper
and grounded disambiguation requires to first develop a significant amount of material;
therefore, this disambiguation is relegated to the aforementioned Section 3.
1.1.1 Alternative representations of the quaternion
The real + imaginary notation {1, i, j, k} is not always convenient for our purposes. Pro-
vided that the algebra (2) is used, a quaternion can be posed as a sum scalar + vector,
Q = q
w
+ q
x
i + q
y
j + q
z
k ⇔ Q = q
w
+ q
v
, (5)
where q
w
is referred to as the real or scalar part, and q
v
= q
x
i + q
y
j + q
z
k = (q
x
, q
y
, q
z
) as
the imaginary or vector part.
1
It can be also defined as an ordered pair scalar-vector
Q = hq
w
, q
v
i . (6)
We mostly represent a quaternion Q as a 4-vector q ,
q ,
q
w
q
v
=
q
w
q
x
q
y
q
z
, (7)
1
Our choice for the (w, x, y, z) subscripts notation comes from the fact that we are interested in the
geometric properties of the quaternion in the 3D Cartesian space. Other texts often use alternative
subscripts such as (0, 1, 2, 3) or (1, i, j, k), perhaps better suited for mathematical interpretations.
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