Some Remarks on the Exponential Map on the Groups SO(n) and SE(n)

Abstract

The problem of describing or determining the image of the exponential map $ \exp :\mathfrak{g}\rightarrow G$ of a Lie group $G$ is important and it has many applications. If the group $G$ is compact, then it is well-known that the exponential map is surjective, hence the exponential image is $G$. In this case the problem is reduced to the computation of the exponential and the formulas strongly depend on the group $G$. In this paper we discuss the generalization of Rodrigues formulas for computing the exponential map of the special orthogonal group ${\rm SO}(n) $, which is compact, and of the special Euclidean group ${\rm SE}(n)$, which is not compact but its exponential map is surjective, in the case $ n\geq 4$.