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Lie Groups for 2D and 3D Transformations

Ethan Eade

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

SO(3)

3D Rotations 3 3D rotation matrix

SE(3)

3D Rigid transformations 6

Linear transformation on

homogeneous 4-vectors

SO(2)

2D Rotations 1 2D rotation matrix

SE(2)

2D Rigid transformations 3

Linear transformation on

homogeneous 3-vectors

Sim(3)

3D Similarity transformations

(rigid motion + scale)

Linear transformation on

homogeneous 4-vectors

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

derived.

1.1 Why use Lie groups for robotics or computer vision?

Many problems in robotics and computer vision involve manipulation and estimation in the 3D geome-

try. Without a coherent and robust framework for representing and working with 3D transformations,

these tasks are onerous and treacherous. Transformations must be composed, inverted, dierentiated

and interpolated. Lie groups and their associated machinery address all of these operations, and do so

in a principled way, so that once intuition is developed, it can be followed with condence.

1.2 Lie algebras and other general properties

Every Lie group has an associated Lie algebra, which is the

tangent space

around the identity element

of the group. That is, the Lie algebra is a vector space generated by dierentiating the group trans-

formations along chosen directions in the space, at the identity transformation. The tangent space

has the same structure at all group elements, though tangent vectors undergo a coordinate transfor-

mation when moved from one tangent space to another. The basis elements of the Lie algebra (and

thus of the tangent space) are called

generators

in this document

All tangent vectors represent linear

combinations of the generators.

Importantly, the tangent space associated with a Lie group provides an optimal space in which to

represent dierential quantities related to the group. For instance, velocities, Jacobians, and covari-

ances of transformations are well-represented in the tangent space around a transformation. This is

the optimal space in which to represent dierential quantities because

•

The tangent space is a vector space with the same dimension as the number of degrees of freedom

of the group transformations

•

The exponential map converts any element of the tangent space

exactly

into a transformation in

the group

•

The adjoint

linearly

and

exactly

transforms tangent vectors from one tangent space to another

The adjoint property is what ensures that the tangent space has the same structure at all points on

the manifold, because a tangent vector can always be transormed back to the tangent space around

the identity.

Each Lie group described below also has a

group action

on 3D space. For instance, 3D rigid transfor-

mations have the action of rotating and translating points. The matrix representations given below

make these actions explicit.

2 SO(3): Rotations in 3D space

2.1 Representation

Elements of the 3D rotation group,

SO(3)

, are represented by 3D rotation matrices. Composition and

inversion in the group correspond to matrix multiplication and inversion. Because rotation matrices

are orthogonal, inversion is equivalent to transposition.

R ∈ SO(3)

(1)

−1

= R

(2)

The Lie algebra,

so(3)

, is the set of

3×3

skew-symmetric matrices. The generators of

so(3)

correspond

to the derivatives of rotation around the each of the standard axes, evaluated at the identity:

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