3D Rotation with Euler Angles

Euler angles are a way to represent a rotation matrix with three rotations around cardinal axes.

In Rotations.jl, there are 12 concrete types for Euler angles.

Proper Euler angles
- RotZXZ, RotXYX, RotYZY, RotZYZ, RotXZX, RotYXY
Tait–Bryan angles
- RotXYZ, RotYZX, RotZXY, RotXZY, RotZYX, RotYXZ

In addition, Rotations.jl provides concrete types that represent rotations in one or two axes.

one axis
- RotX, RotY, RotZ
two axes
- RotXY, RotYZ, RotZX, RotXZ, RotZY, RotYX

Rotation around one axis

\[\begin{aligned} R_{x}(\alpha) &= \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos(\alpha) & -\sin(\alpha) \\ 0 & \sin(\alpha) & \cos(\alpha) \\ \end{pmatrix}, \\ R_{y}(\alpha) &= \begin{pmatrix} \cos(\alpha) & 0 & \sin(\alpha) \\ 0 & 1 & 0 \\ -\sin(\alpha) & 0 & \cos(\alpha) \\ \end{pmatrix}, \\ R_{z}(\alpha) &= \begin{pmatrix} \cos(\alpha) & -\sin(\alpha) & 0\\ \sin(\alpha) & \cos(\alpha) & 0\\ 0 & 0 & 1 \end{pmatrix} \end{aligned}\]

example

Here's an example for RotZ:

julia> α = 1.2  # Rotation angle1.2
julia> R = RotZ(α)3×3 RotZ{Float64} with indices SOneTo(3)×SOneTo(3)(1.2):
 0.362358  -0.932039  0.0
 0.932039   0.362358  0.0
 0.0        0.0       1.0
julia> Q = [cos(α) -sin(α) 0
             sin(α)  cos(α) 0
             0       0      1]
       # These matrices are equal3×3 Matrix{Float64}:
 0.362358  -0.932039  0.0
 0.932039   0.362358  0.0
 0.0        0.0       1.0
julia> R == Qtrue

And more examples for RotX and RotY:

julia> # These matrices are equal
       RotX(α) == [1 0       0
                   0 cos(α) -sin(α)
                   0 sin(α)  cos(α)]
       # These matrices are equaltrue
julia> RotY(α) == [cos(α) 0 sin(α)
                   0      1 0
                  -sin(α) 0 cos(α)]true

Rotation around two axes

\[\begin{aligned} R_{xy}(\alpha,\beta) &= R_{x}(\alpha)R_{y}(\beta), & R_{yx}(\alpha,\beta) &= R_{y}(\alpha)R_{x}(\beta), \\ % R_{yz}(\alpha,\beta) &= R_{y}(\alpha)R_{z}(\beta), & R_{zy}(\alpha,\beta) &= R_{z}(\alpha)R_{y}(\beta), \\ % R_{zx}(\alpha,\beta) &= R_{z}(\alpha)R_{x}(\beta), & R_{xz}(\alpha,\beta) &= R_{x}(\alpha)R_{z}(\beta) \end{aligned}\]

example

julia> α, β = 1.2, 4.7  # Rotation angles(1.2, 4.7)
julia> RotX(α) * RotY(β)3×3 RotXY{Float64} with indices SOneTo(3)×SOneTo(3)(1.2, 4.7):
 -0.0123887  0.0       -0.999923
 -0.931968   0.362358   0.0115467
  0.36233    0.932039  -0.00448913
julia> RotXY(α, β)
       # These matrices are equal3×3 RotXY{Float64} with indices SOneTo(3)×SOneTo(3)(1.2, 4.7):
 -0.0123887  0.0       -0.999923
 -0.931968   0.362358   0.0115467
  0.36233    0.932039  -0.00448913
julia> RotX(α) * RotY(β) == RotXY(α, β)true

Rotation around three axes (Euler Angles)

Proper Euler angles

\[\begin{aligned} R_{xyx}(\alpha, \beta, \gamma) &= R_x(\alpha) R_y(\beta) R_x(\gamma), & R_{yxy}(\alpha, \beta, \gamma) &= R_y(\alpha) R_x(\beta) R_y(\gamma), \\ R_{yzy}(\alpha, \beta, \gamma) &= R_y(\alpha) R_z(\beta) R_y(\gamma), & R_{zyz}(\alpha, \beta, \gamma) &= R_z(\alpha) R_y(\beta) R_z(\gamma), \\ R_{zxz}(\alpha, \beta, \gamma) &= R_z(\alpha) R_x(\beta) R_z(\gamma), & R_{xzx}(\alpha, \beta, \gamma) &= R_x(\alpha) R_z(\beta) R_x(\gamma) \end{aligned}\]

Tait–Bryan angles

\[\begin{aligned} R_{xyz}(\alpha, \beta, \gamma) &= R_x(\alpha) R_y(\beta) R_z(\gamma), & R_{yxz}(\alpha, \beta, \gamma) &= R_y(\alpha) R_x(\beta) R_z(\gamma), \\ R_{yzx}(\alpha, \beta, \gamma) &= R_y(\alpha) R_z(\beta) R_x(\gamma), & R_{zyx}(\alpha, \beta, \gamma) &= R_z(\alpha) R_y(\beta) R_x(\gamma), \\ R_{zxy}(\alpha, \beta, \gamma) &= R_z(\alpha) R_x(\beta) R_y(\gamma), & R_{xzy}(\alpha, \beta, \gamma) &= R_x(\alpha) R_z(\beta) R_y(\gamma) \end{aligned}\]

example

julia> α, β, γ = 1.2, 4.7, -0.4  # Rotation angles(1.2, 4.7, -0.4)
julia> RotYXZ(α, β, γ)3×3 RotYXZ{Float64} with indices SOneTo(3)×SOneTo(3)(1.2, 4.7, -0.4):
  0.696679    -0.71729    -0.0115467
  0.00482437  -0.0114107   0.999923
 -0.717367    -0.696681   -0.00448913
julia> RotY(α)*RotX(β)*RotZ(γ)
       # These matrices are equal3×3 RotYXZ{Float64} with indices SOneTo(3)×SOneTo(3)(1.2, 4.7, -0.4):
  0.696679    -0.71729    -0.0115467
  0.00482437  -0.0114107   0.999923
 -0.717367    -0.696681   -0.00448913
julia> RotYXZ(α, β, γ) == RotY(α)*RotX(β)*RotZ(γ)true