Angle2d seems superfluous

[cafaxo]

It seems that Angle2d represents a 2D rotation by an angle.
This seems confusing to me: This representation is basically the same as the Lie algebra representation by a 2x2 skew-symmetric matrix.
So if I want to represent a 2D rotation "lazily" by an angle, I would use the proper Lie algebra type RotationGenerator{2,T}.
What is the point of having Angle2d?

When I exponentiate the RotationGenerator{2,T} I would expect to receive the actual rotation matrix.
Having to apply an additional RotMatrix is not nice when writing generic code that works for multiple dimensions.

Edit: It seems that 3D rotations also behave in the same way with first producing a RotationVec when exponentiating an element from the Lie algebra.
It just seems a bit opaque to me that the actual work is delayed to the RotMatrix constructor and does not happen in the exp.

[cafaxo]

After looking at this some more, I understand why it is done this way. (exp does not make the decision how the rotation is represented; and there are various representations of 3D rotations available).

[hyrodium]

there are various representations of 3D rotations available

Yeah, that's right.

Here's a draft comment that I was trying to send.

---

What is the point of having Angle2d?

For more performance on 2D rotations. (See #14, #89 and docs)

julia> using Rotations, StaticArrays, BenchmarkTools

julia> subtypes(Rotation{2})
2-element Vector{Any}:
 Angle2d
 RotMatrix{2}

julia> r1 = Angle2d(3.0)
2×2 Angle2d{Float64} with indices SOneTo(2)×SOneTo(2)(3.0):
 -0.989992  -0.14112
  0.14112   -0.989992

julia> r2 = RotMatrix{2}(3.0)
2×2 RotMatrix2{Float64} with indices SOneTo(2)×SOneTo(2):
 -0.989992  -0.14112
  0.14112   -0.989992

julia> @btime r1^3
  16.464 ns (1 allocation: 16 bytes)
2×2 Angle2d{Float64} with indices SOneTo(2)×SOneTo(2)(9.0):
 -0.91113   -0.412118
  0.412118  -0.91113

julia> @btime r2^3
  68.839 ns (1 allocation: 48 bytes)
2×2 RotMatrix2{Float64} with indices SOneTo(2)×SOneTo(2):
 -0.91113   -0.412118
  0.412118  -0.91113

I would use the proper Lie algebra type RotationGenerator{2,T}.

You can also use Angle2dGenerator type.

julia> g = Angle2dGenerator(3.0)
2×2 Angle2dGenerator{Float64} with indices SOneTo(2)×SOneTo(2)(3.0):
 0.0  -3.0
 3.0   0.0

julia> exp(g)
2×2 Angle2d{Float64} with indices SOneTo(2)×SOneTo(2)(3.0):
 -0.989992  -0.14112
  0.14112   -0.989992

It just seems a bit opaque to me that the actual work is delayed to the RotMatrix constructor and does not happen in the exp.

There are many ways to represent a rotations: Euler angles, rotation around an axis, quaternions, etc. Rotations.jl has types for each rotation representation.

julia> subtypes(Rotation{3})
27-element Vector{Any}:
 AngleAxis
 MRP
 QuatRotation
 RodriguesParam
 RotMatrix{3}
 RotX
 RotXY
 RotXYX
 RotXYZ
 RotXZ
 RotXZX
 RotXZY
 RotY
 RotYX
 RotYXY
 RotYXZ
 RotYZ
 RotYZX
 RotYZY
 RotZ
 RotZX
 RotZXY
 RotZXZ
 RotZY
 RotZYX
 RotZYZ
 RotationVec

[cafaxo]

Thanks for elaborating on this!