these tasks are onerous and treacherous. Transformations must be composed, inverted, dierentiated
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 condence.
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 dierentiating 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 dierential 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 dierential 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)
R
−1
= R
T
(2)
2
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