Add RotationGenerator and its subtypes

[hyrodium]

This PR solves #30, #31, #126 and #128. (2021/11/20 EDIT: quaternion conversions will be supported in the other PR.)

julia> using Rotations

julia> m = rand(3,3)
3×3 Matrix{Float64}:
 0.992789  0.0478161  0.639771
 0.826576  0.89928    0.368333
 0.100041  0.986343   0.174436

julia> s = InfinitesimalRotMatrix{3}(m - m')
3×3 InfinitesimalRotMatrix{3, Float64, 9} with indices SOneTo(3)×SOneTo(3):
  0.0      -0.77876   0.53973
  0.77876   0.0      -0.61801
 -0.53973   0.61801   0.0

julia> exp(s)
3×3 RotationVec{Float64} with indices SOneTo(3)×SOneTo(3)(0.61801, 0.53973, 0.77876):
  0.596987  -0.473236   0.647807
  0.772707   0.556304  -0.305697
 -0.215711   0.683062   0.697779

julia> log(exp(s))
3×3 InfinitesimalRotationVec{Float64} with indices SOneTo(3)×SOneTo(3)(0.61801, 0.53973, 0.77876):
  0.0      -0.77876   0.53973
  0.77876   0.0      -0.61801
 -0.53973   0.61801   0.0

julia> log(exp(s)) + s
3×3 InfinitesimalRotMatrix{3, Float64, 9} with indices SOneTo(3)×SOneTo(3):
  0.0      -1.55752   1.07946
  1.55752   0.0      -1.23602
 -1.07946   1.23602   0.0

julia> rotation_angle(exp(log(exp(s)) + s))
2.2624870696189996

julia> rotation_angle(exp(s))
1.1312435348094998

julia> InfinitesimalRotationVec(s)
3×3 InfinitesimalRotationVec{Float64} with indices SOneTo(3)×SOneTo(3)(0.61801, 0.53973, 0.77876):
  0.0      -0.77876   0.53973
  0.77876   0.0      -0.61801
 -0.53973   0.61801   0.0

[codecov]

Codecov Report

Merging #203 (b55c9ee) into master (89e4dd4) will increase coverage by 0.83%.
The diff coverage is 93.61%.

@@            Coverage Diff             @@
##           master     #203      +/-   ##
==========================================
+ Coverage   86.74%   87.57%   +0.83%     
==========================================
  Files          14       15       +1     
  Lines        1403     1529     +126     
==========================================
+ Hits         1217     1339     +122     
- Misses        186      190       +4     

Impacted Files	Coverage Δ
src/core_types.jl	93.89% <ø> (+3.45%)	⬆️
src/logexp.jl	81.81% <81.81%> (ø)
src/rotation_generator.jl	95.79% <95.79%> (ø)

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[andyferris]

Oh cool. :)

I would still avoid the term "infinitesimal rotation". Julia can represent infinitesimal rotations already, but it's not these. Here's one, with dual numbers:

using DualNumbers

infinitesimal_rotation = [
    1.0 Dual(0.0, 0.5);
    Dual(0.0, -0.5) 1.0
]

What you have in this PR is a rotation generator, so if we aren't going to refer to the matrix structure then we may as well call it by its real name. Note the generator for the above an infinitesimal generator:

infinitesimal_generator = [
    0.0 Dual(0.0, 0.5);
    Dual(0.0, -0.5) 0.0
]

It's easy to figure this out using just exp(ϵ) ≈ I + ϵ.

What doesn't quite work out-of-the-box in Julia is exp and log with Dual matrices, but mathematically the following should hold:

exp(infinitesimal_generator) == infinitesimal_rotation

Hopefully with your work here we'll be able to get all these relationships working even with autodifferentiation on the parameters of the generator! :)

[hyrodium]

I would still avoid the term "infinitesimal rotation". Julia can represent infinitesimal rotations already, but it's not these. Here's one, with dual numbers:

Ah, I didn't quite understand the meaning of inifinitesimal rotation.
Should we just rename the skew-symmetric matrix types like this?

• InfinitesimalRotation -> RotationGenerator
• InfinitesimalRotMatrix -> RotMatrixGenerator
• InfinitesimalRotationVec -> RotationVecGenerator
• InfinitesimalAngle2c -> Angle2dGenerator

Hopefully with your work here we'll be able to get all these relationships working even with autodifferentiation on the parameters of the generator! :)

Yeah, I'll try to do that! 😄

[andyferris]

Should we just rename the skew-symmetric matrix types like this?

Yes - I think that is perfect :)

[andyferris]

Awesome.

I had questions about the method defined for promote_op (and perhaps whatever you are trying to acheive there could be tested, too?) but otherwise this looks good to me.

This PR fills a massive gap, and I'm sure the community will appreciate it. Thanks @hyrodium!

[andyferris]

Is this right? The operator is not defined. Did you mean promote_rule?

[hyrodium]

Ah, I was just referring to src/core_types.jl.
0ccc876#diff-55fa94d9eb6d180ed431c8aae0ad29c7100a0ad88cbd35dfc81239882a614e8dR39

[hyrodium]

I've tried renaming promote_op -> promote_rule and isleaftype -> isconcretetype, but it doesn't work well here.
These seems unnecessary, so I've dropped both promote_op definitions from src/core_types.jl and src/rotation_generator.jl.