The manner in which complex numbers were used to represent 2D
rotations will now be adapted to using quaternions to represent 3D
rotations. Let
represent the set of *quaternions*, in which
each quaternion,
, is represented as
, and
. A quaternion can be considered as a
four-dimensional vector. The symbols , , and are used to
denote three ``imaginary'' components of the quaternion. The
following relationships are defined:
,
from which it follows that , , and . Using
these, multiplication of two quaternions,
and
, can be derived to
obtain
, in which

Using this operation, it can be shown that is a group with respect to quaternion multiplication. Note, however, that the multiplication is not commutative! This is also true of 3D rotations; there must be a good reason.

For convenience, quaternion multiplication can be expressed in terms of vector multiplications, a dot product, and a cross product. Let be a three-dimensional vector that represents the final three quaternion components. The first component of is . The final three components are given by the three-dimensional vector .

In the same way that *unit* complex numbers were needed for , *unit
quaternions* are needed for , which means that
is
restricted to quaternions for which
. Note
that this forms a subgroup because the multiplication of unit
quaternions yields a unit quaternion, and the other group axioms hold.

The next step is to describe a mapping from unit quaternions to . Let the unit quaternion map to the matrix

which can be verified as orthogonal and . Therefore, it belongs to . It is not shown here, but it conveniently turns out that represents the rotation shown in Figure 4.9, by making the assignment

(4.21) |

Unfortunately, this representation is not unique. It can be verified in (4.20) that . A nice geometric interpretation is given in Figure 4.10. The quaternions and represent the same rotation because a rotation of about the direction is equivalent to a rotation of about the direction . Consider the quaternion representation of the second expression of rotation with respect to the first. The real part is

(4.22) |

The , , and components are

(4.23) |

The quaternion has been constructed. Thus, and represent the same rotation. Luckily, this is the only problem, and the mapping given by (4.20) is two-to-one from the set of unit quaternions to .

This can be fixed by the identification trick. Note that the set of unit quaternions is homeomorphic to because of the constraint . The algebraic properties of quaternions are not relevant at this point. Just imagine each as an element of , and the constraint forces the points to lie on . Using identification, declare for all unit quaternions. This means that the antipodal points of are identified. Recall from the end of Section 4.1.2 that when antipodal points are identified, . Hence, , which can be considered as the set of all lines through the origin of , but this is hard to visualize. The representation of in Figure 4.5 can be extended to . Start with , and make three different kinds of identifications, one for each pair of opposite cube faces, and add all of the points to the manifold. For each kind of identification a twist needs to be made (without the twist, would be obtained). For example, in the direction, let for all .

One way to force uniqueness of rotations is to require staying in the ``upper half'' of . For example, require that , as long as the boundary case of is handled properly because of antipodal points at the equator of . If , then require that . However, if , then require that because points such as and are the same rotation. Finally, if , then only is allowed. If such restrictions are made, it is important, however, to remember the connectivity of . If a path travels across the equator of , it must be mapped to the appropriate place in the ``northern hemisphere.'' At the instant it hits the equator, it must move to the antipodal point. These concepts are much easier to visualize if you remove a dimension and imagine them for , as described at the end of Section 4.1.2.

Steven M LaValle 2012-04-20