The exponential map for the group of similarity transformations and applications to motion interpolation

In this paper, we explore the exponential map and its inverse, the logarithm map, for the group SIM(n) of similarity transformations in ℝⁿ which are the composition of a rotation, a translation and a uniform scaling. We give a formula for the exponential map and we prove that it is surjective. We give an explicit formula for the case of n = 3 and show how to efficiently compute the logarithmic map. As an application, we use these algorithms to perform motion interpolation. Given a sequence of similarity transformations, we compute a sequence of logarithms, then fit a cubic spline that interpolates the logarithms and finally, we compute the interpolating curve in SIM(3).