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Quaternions and spatial rotation 135
Conversion to and from the matrix representation
From a quaternion to an orthogonal matrix
The orthogonal matrix corresponding to a rotation by the unit quaternion (with |z| = 1) is
given by
From an orthogonal matrix to a quaternion
Finding the quaternion that corresponds to a rotation matrix can be numerically
unstable if the trace (sum of the diagonal elements) of the rotation matrix is zero or very small. A robust method is to
choose the diagonal element with the largest value. Let uvw be an even permutation of xyz (i.e. xyz, yzx or zxy).
The value
will be a real number because . If r is zero the matrix is the identity matrix, and the
quaternion must be the identity quaternion (1, 0, 0, 0). Otherwise the quaternion can be calculated as follows: [3]
Beware the vector convention: There are two conventions for rotation matrices: one assumes row vectors on the left;
the other assumes column vectors on the right; the two conventions generate matrices that are the transpose of each
other. The above matrix assumes column vectors on the right. In general, a matrix for vertex transpose is ambiguous
unless the vector convention is also mentioned. Historically, the column-on-the-right convention comes from
mathematics and classical mechanics, whereas row-vector-on-the-left comes from computer graphics, where
typesetting row vectors was easier back in the early days.
(Compare the equivalent general formula for a 3 × 3 rotation matrix in terms of the axis and the angle.)
Fitting quaternions
The above section described how to recover a quaternion q from a 3 × 3 rotation matrix Q. Suppose, however, that
we have some matrix Q that is not a pure rotation — due to round-off errors, for example — and we wish to find the
quaternion q that most accurately represents Q. In that case we construct a symmetric 4×4 matrix
and find the eigenvector (x,y,z,w) corresponding to the largest eigenvalue (that value will be 1 if and only if Q is a
pure rotation). The quaternion so obtained will correspond to the rotation closest to the original matrix Q [4]