LieTypes.jl

Julia types for Lie groups.

The main goal is for the provided types to allow the writing of Lie-group-agnostic code.

The interface consists of only 4 methods¹:

• *(::T, ::T)::T where {T<:LieGroup}: element multiplication.

• inv(::T)::T where {T<:LieGroup}: element inverse.

• exp(::Type{T}, ::Array)::T where {T<:LieGroup}: exponemential map.

• log(<:LieGroup)::Array: logarithm.

Each concrete type provides their own constructors and selectors, as well as some encoding of the Lie algebra (always represented as an Array).

Currently 6 types are provided (LieScalar, LieVector, SO2, SE2, SO3, and SE3), all related to rigid-body transforms.

Support Types

3 support types are provided:

• DualComplex, which implements what is sometimes referred as "dual complex numbers" or "planar quaternions" (see, for instance, Applications of dual quaternions to 2D geometry). This type is used as the underlying representation of SE2, and as a valid interface (via the se2_from_dual_complex constructor and the dual_complex selector).

• Quaternion, which implements quaternions. This type is used as the underlying representation of SO3, and as a valid interface (via the so3_from_quaternion constructor and quaternion selector).

• Dual, which implements dual numbers. The type Dual{Quaternion{Float64}} is used as the underlying representation of SE3, and as a valid interface (via the se3_from_dual_quaternion constructor and dual_quaternion selector).

¹ The provided type annotations are only valid for non-parametric subtypes of LieGroup.