Fast closest logarithm algorithm in the special orthogonal group
Adrien Escande · IMA Journal of Numerical Analysis · 2015
For interpolating between elements of |${\rm SO}{(n)}$|, it is attractive to work in |$n$|, passing from one space to the other via the exponential map. However, the logarithm is a multi-valued map and the choice of a particular image affects the quality of the interpolation. In this paper, we propose a fast and accurate algorithm to compute the image that seems the most appropriate for interpolation: given |$Q \in {\rm SO}{(n)}$| and |$A \in n$|, our algorithm returns the logarithm of |$Q$| which is the closest to |$A$|, under minimal conditions on |$Q$|. We carefully study the mathematical properties of our problem to establish the algorithm, discuss its implementation and demonstrate its efficiency.