How to use rotation objects to compute coordinates of a given point?

[juliohm]

First off, I really appreciate your work @ronisbr, it is super useful. I am doing some research on how to operate with coordinate transformations efficiently and really liked the approach of this package. I wonder if you have an example of how these reference frames are used to compute coordinates of a point?

Say for example that we have a point with coordinates (1,2,3) with respect to the canonical Euclidean basis (i.e. (1,0,0), (0,1,0), (0,0,1)), which I understand can be easily constructed with R = DCM(I). For this rotation object, we can simply do R*[1,2,3] to retrieve the coordinates of the point. Now, suppose we are on a different frame of reference like R = EulerAngles(0.5,0.5,0.5), how do you compute the coordinates of the point with respect to this frame? Do we need to convert to DCM first?

[ronisbr]

Hi @juliohm !

First off, I really appreciate your work @ronisbr, it is super useful.

Nice!! I am glad it is being useful :)

The idea here is to provide some tools to rotate reference frames. It is assumed that all frames are right-handed and orthonormal.

Let's suppose you have an origin frame, which is called o, and a destination frame, which is called d. We know, for example, that those frames are related by the following rotations (Euler angles):

• 0.3 rad in X;
• 0.1 rad in Y;
• 1.3 rad in Z.

Hence, we can construct, for example, a DCM that rotates the o reference frame into alignment to d:

julia> Ddo = angle_to_dcm(0.3, 0.1, 1.3, :XYZ)
3×3 StaticArrays.SArray{Tuple{3,3},Float64,2,9} with indices SOneTo(3)×SOneTo(3):
  0.266162    0.928414  0.259238
 -0.958744    0.227124  0.17095
  0.0998334  -0.294044  0.950564

Now, if a vector v has the representation [1,2,3] in the o reference frame, then its representation in the d reference frame can be found by:

julia> v_o = [1,2,3]
3-element Array{Int64,1}:
 1
 2
 3

julia> v_d = Ddo*v_o
3-element StaticArrays.SArray{Tuple{3},Float64,1,3} with indices SOneTo(3):
 2.900705997639297
 0.008353646752887522
 2.3634371013093065

The same can be accomplished by Quaternions, which is another method to represent rotations:

julia> qdo = angle_to_quat(0.3, 0.1, 1.3, :XYZ)
Quaternion{Float64}:
  + 0.7816408973671287 + 0.14872367635093683.i - 0.05098406742924803.j + 0.6035887677252959.k

julia> vect(qdo\v_o*qdo)
3-element StaticArrays.SArray{Tuple{3},Float64,1,3} with indices SOneTo(3):
 2.9007059976392973
 0.008353646752887467
 2.3634371013093065

[juliohm]

That is really nice @ronisbr , thanks for clarifying the workflow. We are adding ReferenceFrameRotations.jl as a dependency in GeoStats.jl: https://github.com/JuliaEarth/GeoStatsBase.jl/pull/55

Closing the issue, and looking forward to playing with the rotations. :+1:

[ronisbr]

Nice!! If you need any help, please, let me know.