Some Computational Properties of Rotation Representations

There are four main parameterizations of the rotation group SO(3). Two of them (rotation angle and axis, and the closely related quaternion components) as well as the matrix form of rotation representation are particularly of interest in computer vision, graphics, and robotics. Some of their computational properties are explored here. Operation counts are given for primitive operations of normalization, conversion, and application of rotations as well as for sequences of rotations. The numerical accuracy of vector rotation calculations is investigated for some common tasks like iterated application of the same or different rotations. The measure of accuracy is taken to be the length and direction of the resulting rotated vector. Some analytical analysis appears, but most conclusions are based on empirical tests at artificially reduced numerical precision