Add more abstract subtypes of `Rotation`?

The problems

type conversion

On the current master branch:

julia> UnitQuaternion(RotX(2π))
3×3 UnitQuaternion{Float64} with indices SOneTo(3)×SOneTo(3)(1.0, -1.22465e-16, 0.0, 0.0):
  1.0  -0.0          0.0
  0.0   1.0          2.44929e-16
 -0.0  -2.44929e-16  1.0

The result should approximately equal to UnitQuaternion(-1,0,0,0), not UnitQuaternion(1,0,0,0). (See https://github.com/JuliaGeometry/Rotations.jl/issues/171#issuecomment-950324745)

power operator ^

On the current master branch:

julia> AngleAxis(2π, 0,0,1)^(1/4)
3×3 AngleAxis{Float64} with indices SOneTo(3)×SOneTo(3)(1.5708, 0.0, 0.0, 1.0):
 2.22045e-16  -1.0          0.0
 1.0           2.22045e-16  0.0
 0.0           0.0          1.0

julia> AngleAxis(0, 0,0,1)^(1/4)
3×3 AngleAxis{Float64} with indices SOneTo(3)×SOneTo(3)(0.0, 0.0, 0.0, 1.0):
 1.0  0.0  0.0
 0.0  1.0  0.0
 0.0  0.0  1.0

julia> AngleAxis(0, 0,0,1) ≈ AngleAxis(2π, 0,0,1)
true

I think this is okay, but it would be nice if it will be easier to understand.

product operator *

On the current master branch:

julia> UnitQuaternion(1.,0.,0.,0.) * MRP(1.,2.,3.)
3×3 RotMatrix3{Float64} with indices SOneTo(3)×SOneTo(3):
  0.537778   0.764444   -0.355556
 -0.622222   0.644444    0.444444
  0.568889  -0.0177778   0.822222

Both UnitQuaternion and MRP are points on SU(2), but the output result is RotMatrix (on SO(3)).
This means the operation drops the information on SU(2).

summary

The subtypes of Rotation are treated as a real matrix, but it is not just a matrix.

How to fix this

Classify the types

There are 27 subtypes of Rotation{3}, and this can be classified as the following.

• Rotation on SO(3)
  • Its parameters are bijective to SO(3), almost everywhere.
  • Should be named RotationSO3 <: Rotation{3}??
  • its subtypes
    • RotMatrix{3}
    • RodriguteParam
• Rotation on SU(2)
  • Its parameters are bijective to SU(2), almost everywhere.
  • Should be named RotationSU2 <: Rotation{3}??
  • its subtypes
    • UnitQuaternion (QuatRotation in the future)
    • MRP
• Path on the rotational space
  • Its can be considered as a path on rotational space (SO(3) or SU(2)).
  • The mapping parameters doesn't have to be injective or surjective.
    • All of its subtypes are not injective. (e.g. RotX(0) = Rot(2π) on SO(3), RotX(0) = Rot(4π) on SU(2))
    • Some of them are surjective (e.g. RotXYZ, AngleAxis)
  • Should be named RotationPath{3} <: Rotation{3}??
  • its subtypes
    • RotX and etc.
    • RotXY and etc.
    • RotXYZ and etc.
    • AngleAxis
    • RotationVec

Methods

With those abstract types (RotationSO3, RotationSU2 and RotationPath), we can update methods like the following:

UnitQuaternion(::RotXYZ)  # This conversion is surjective on SU(2)
MRP(::RotXYZ)  # This conversion is surjective on SU(2)
RodriguesParam(::MRP)  # This conversion is surjective on SO(3)
MRP(::RodriguesParam)  # This conversion is injective on SU(2), and chosen principal value (PR#97)

*(::RotationSO3,::RotationSO3) isa RotationSO3
*(::RotationSU2,::RotationSU2) isa RotationSU2
*(::RotationSO3,::RotationSU2) isa RotationSO3  # This is for associativity, (pq)r = p(qr)
*(::RotationPath,::RotationPath) isa Union{RotationPath, RotationSU2}  # If possible, should be RotationPath. (e.g. RotX(1)*RotY(2))  In other cases, should be RotationSU2 because SU(2) is a simply-connected.
*(::RotationPath,::RotationSO3) isa RotationSO3
*(::RotationPath,::RotationSU2) isa RotationSO2

0 ≤ rotation_angle(::RotationSO3) < π
0 ≤ rotation_angle(::RotationSU2) < 2π
-∞ < rotation_angle(::RotationPath) < +∞

exp(::InfinitesimalRotation{3}) isa RotationVec  # Note that RotationVec <: RotationPath

Questions

• Does this sounds good?
• If so, are there any suggestions for naming types?
  • I think there would be better namings than RotationSO3 and RotationSU2.

Start by reproducing the UnitQuaternion conversion, power, and mixed-product examples from the issue on the current master branch. Then review the existing Rotation subtype hierarchy and methods; done would require an agreed design for RotationSO3, RotationSU2, and RotationPath, including naming and dispatch behavior, but no files or tests are identified here.