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Rotations are not always performed about the principal coordinate axes. We are often interested in a rotation about an arbitrary axis in space. This provides both a convenient way to describe rotations, and an alternative parameterization for rotation matrices. Let k = (kx, ky, kz), expressed in the frame o₀x₀y₀z₀, be a unit vector defining an axis. We wish to derive the rotation matrix representing a rotation of θ about this axis.

There are several ways in which the matrix can be derived. One approach is to note that the rotational transformation will bring the world z-axis into alignment with the vector k. Therefore, a rotation about the axis k can be computed using a similarity transformation as

(2.40)

(2.41)

From Figure 2.12 we see that

(2.42)

(2.43)

Note that the final two equations follow from the fact that k is a unit vector. Substituting Equations (2.42) and (2.43) into Equation (2.41), we obtain after some lengthy calculation (Problem 2–17)

(2.44)

where v_θ = vers θ = 1 − c_θ.

Figure 2.12 Rotation about an arbitrary axis.

In fact, any rotation matrix can be represented by a single rotation about a suitable axis in space by a suitable angle,

(2.45)

where k is a unit vector defining the axis of rotation, and θ is the angle of rotation about k. The pair (k, θ) is called the axis-angle representation of . Given an arbitrary rotation matrix with components rij, the equivalent angle θ and equivalent axis k are given by the expressions

and

(2.46)

These equations can be obtained by direct manipulation of the entries of the matrix given in Equation (2.44). The axis-angle representation is not unique since a rotation of − θ about − k is the same as a rotation of θ about k, that is,

(2.47)

If θ = 0 then is the identity matrix and the axis of rotation is undefined.

Example 2.9.

Suppose is generated by a rotation of 90° about z₀ followed by a rotation of 30° about y₀ followed by a rotation of 60° about x₀. Then

(2.48)

We see that and hence the equivalent angle is given by Equation (2.46) as

(2.49)

The equivalent axis is given from Equation (2.46) as

(2.50)

The above axis-angle representation characterizes a given rotation by four quantities, namely the three components of the equivalent axis k and the equivalent angle θ. However, since the equivalent axis k is given as a unit vector only two of its components are independent. The third is constrained by the condition that k is of unit length. Therefore, only three independent quantities are required in this representation of a rotation . We can represent the equivalent axis-angle by a single vector r as

(2.51)

Note, since k is a unit vector, that the length of the vector r is the equivalent angle θ and the direction of r is the equivalent axis k.

One should be careful to note that the representation in Equation (2.51) does not mean that two axis-angle representations may be combined using standard rules of vector algebra, as doing so would imply that rotations commute which, as we have seen, is not true in general.