0.4 Molecular distance measures  (Page 5/6)

Quaternions and three-dimensional rotations

A number of different methods exist for denoting rotations of rigid objects in three-dimensional space. These are introduced in a module on protein kinematics . Unit quaternions represent a rotation of an angle around an arbitrary axis. A rotation by the angle theta about an axis represented by the unit vector v = [x, y, z] is represented by a unit quaternion:

Like rotation matrices, quaternions may be composed with each other via multiplication. The major advantage of the quaternion representation is that it is more robust to numerical instability than orthonormal matrices. Numerical instability results from the fact that, because computers use a finite number of bits to represent real numbers, most real numbers are actually represented by the nearest number the computer is capable of representing. Over a series of floating point operations, the error caused by this inexact representation accumulates, quite rapidly in the case of repeated multiplications and divisions. In manipulating orthonormal transformation matrices, this can result in matrices that are no longer orthonormal, and therefore not valid rigid transformations. Finding the "nearest" orthonormal matrix to an arbitrary matrix is not a well-defined problem. Unit-length quaternions can accumulate the same kind of a numerical error as rotation matrices, but in the case of quaternions, finding the nearest unit-length quaternion to an arbitrary quaternion is well defined. Additionally, because quaternions correspond more directly to the axis-angle representation of three-dimensional rotations, it could be argued that they have a more intuitive interpretation than rotation matrices. Quaternions, with four parameters, are also more memory efficient than 3x3 matrices. For all of these reasons, quaternions are currently the preferred representation for three-dimensional rotations in most modeling applications.

Vectors can be represented as purely imaginary quaternions, that is, quaternions whose scalar component is 0. The quaternion corresponding to the vector v = [x, y, z] is q = xi + yj + zk.

We can perform rotation of a vector in quaternion notation as follows:

Optimal alignment with quaternions

The method presented here is from Berthold K. P. Holm, "Closed-form solution of absolute orientation using unit quaternions." Journal of the Optical Society of America A, 4:629-642.

The alignment problem may be stated as follows:

• We have two sets of points (atoms) A and B for which we wish to find an optimal alignment, defined as the alignment for which the root mean square difference between each point in A and its corresponding point in B is minimized.
• We know which point in A corresponds to which point in B. This is necessary for any RMSD-based method.