Euler Angle and Axis

The Euler angle and axis representation is defined by the following immutable structure:

struct EulerAngleAxis{T}
    a::T
    v::SVector{3,T}
end

in which a is the Euler Angle and v is a unitary vector aligned with the Euler axis.

The constructor for this structure is:

function EulerAngleAxis(a::T1, v::AbstractVector{T2}) where {T1,T2}

in which a EulerAngleAxis with angle a [rad] and vector v will be created. Notice that the type of the returned structure will be selected according to the input types T1 and T2. Furthermore, the vector v will not be normalized.

julia> EulerAngleAxis(1, [1, 1, 1])
EulerAngleAxis{Int64}:
  Euler angle : 1 rad  (57.2958°)
  Euler axis  : [1, 1, 1]

julia> EulerAngleAxis(1.f0, [1, 1, 1])
EulerAngleAxis{Float32}:
  Euler angle : 1.0 rad  (57.2958°)
  Euler axis  : [1.0, 1.0, 1.0]

julia> EulerAngleAxis(1, [1, 1, 1.f0])
EulerAngleAxis{Float32}:
  Euler angle : 1.0 rad  (57.2958°)
  Euler axis  : [1.0, 1.0, 1.0]

julia> EulerAngleAxis(1.0, [1, 1, 1])
EulerAngleAxis{Float64}:
  Euler angle : 1.0 rad  (57.2958°)
  Euler axis  : [1.0, 1.0, 1.0]

Operations

Multiplication

The multiplication of two Euler angle and axis sets is defined here as the composition of the rotations. Let $\Theta_1$ and $\Theta_2$ be two Euler angle and axis sets (instances of the structure EulerAngleAxis). Thus, the operation:

\[\Theta_{2,1} = \Theta_2 \cdot \Theta_1\]

will return a new set of Euler angle and axis $\Theta_{2,1}$ that represents the composed rotation of $\Theta_1$ followed by $\Theta_2$. By convention, the Euler angle of the result will always be in the interval $[0, \pi]$ rad.

julia> ea1 = EulerAngleAxis(30 * pi / 180, [1.0, 0.0, 0.0])
EulerAngleAxis{Float64}:
  Euler angle : 0.523599 rad  (30.0°)
  Euler axis  : [1.0, 0.0, 0.0]

julia> ea2 = EulerAngleAxis(60 * pi / 180, [1.0, 0.0, 0.0])
EulerAngleAxis{Float64}:
  Euler angle : 1.0472 rad  (60.0°)
  Euler axis  : [1.0, 0.0, 0.0]

julia> ea2 * ea1
EulerAngleAxis{Float64}:
  Euler angle : 1.5708 rad  (90.0°)
  Euler axis  : [1.0, 0.0, 0.0]

Inversion

The inv function applied to Euler angle and axis will return the inverse rotation. Hence, if the Euler angle is a and the Euler axis is aligned with the unitary vector v, then it will return a as the Euler angle and -v as the Euler axis. By convention, the Euler angle of the result will always be in the interval $[0, \pi]$ rad.

julia> ea = EulerAngleAxis(1.3, [1.0, 0, 0]);

julia> inv(ea)
EulerAngleAxis{Float64}:
  Euler angle : 1.3 rad  (74.4845°)
  Euler axis  : [-1.0, -0.0, -0.0]

julia> ea = EulerAngleAxis(-π, [sqrt(3), sqrt(3), sqrt(3)]);

julia> inv(ea)
EulerAngleAxis{Float64}:
  Euler angle : 3.14159 rad  (180.0°)
  Euler axis  : [-1.73205, -1.73205, -1.73205]

julia> ea = EulerAngleAxis(-3π / 2, [sqrt(3), sqrt(3), sqrt(3)]);

julia> inv(ea)
EulerAngleAxis{Float64}:
  Euler angle : 1.5708 rad  (90.0°)
  Euler axis  : [-1.73205, -1.73205, -1.73205]