1 Introduction

This document derives useful formulae for working with the Lie groups that represent transformations in 2D and 3D space. A Lie group is a topological group that is also a smooth manifold, with some other nice properties. Associated with every Lie group is a Lie algebra, which is a vector space discussed below. Importantly, a Lie group and its Lie algebra are intimately related, allowing calculations in one to be mapped usefully into the other.

This document does not give a rigorous introduction to Lie groups, nor does it discuss all of the mathematical details of Lie groups in general. It does attempt to provide enough information that the Lie groups representing spatial transformations can be employed usefully in robotics and computer vision.

Here are the Lie groups that this document addresses:

Group	Description	Dim.	Matrix Representation
$\mathrm{SO}(3)$	3D Rotations	3	3D rotation matrix
$\mathrm{SE}(3)$	3D Rigid transformations	6	Linear transformation on homogeneous 4-vectors
$\mathrm{SO(2)}$	2D Rotations	1	2D rotation matrix
$\mathrm{SE(2)}$	2D Rigid transformations	3	Linear transformation on homogeneous 3-vectors
$\mathrm{Sim}(3)$	3D Similarity transformations (rigid motion + scale)	7	Linear transformation on homogeneous 4-vectors

For each of these groups, the representation is described, and the exponential map and adjoint are derived.