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

Spyridon Leonardos; Christine Allen-Blanchette; Jean Gallier

ICRA 2015poster4 citations

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

Spyridon Leonardos, Christine Allen-Blanchette, Jean Gallier

Abstract

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).

BibTeX

@inproceedings{icra2015_theexponentialma,
  title = {The exponential map for the group of similarity transformations and applications to motion interpolation},
  author = {Spyridon Leonardos and Christine Allen-Blanchette and Jean Gallier},
  booktitle = {ICRA 2015},
  year = {2015}
}