Quaternions

Submodule of DeformableBodies.jl implementing quaternion algebra.

Quaternions are a 4-dimensional normed division algebra which extend the complex numbers. They may be used as a representation of rotations on 3-dimensional space.

Exports

• Quaternion
• axis
• axistoquaternion
• components
• imagq
• matrixtoquaternion
• normalize
• quaterniontomatrix
• rotate

Quaternion{T<:Real} <: Number

Quaternion type with components of type T.

This type overloads all the arithmetic operations as well as the methods defined for Complex numbers that still make sense for Quaternions.

components(q)

Return an array with the components of a Quaternion.

Example

julia> components(Quaternion(1.0, 2.0, 3.0, 4.0))
4-element Array{Float64,1}:
 1.0
 2.0
 3.0
 4.0

imagq(q)

Return imaginary part of Quaternion as a Quaternion with no real part.

Examples

julia> a = Quaternion(1,2,3,4)
1 + 2i + 3j + 4k

julia> imagq(a)
0 + 2i + 3j + 4k

axis(q)

Return the unit vector on the direction of the imaginary part of a Quaternion.

Examples

julia> Quaternion(10,1,1,0.5)
10.0 + 1.0i + 1.0j + 0.5k

julia> axis(Quaternion(10,1,1,0.5))
3-element Array{Float64,1}:
 0.6666666666666666
 0.6666666666666666
 0.3333333333333333

normalize(q)

Return a Quaternion with the same direction as q but unit norm.

Examples

julia> q = Quaternion([1., 1., 1., 1.])
1.0 + 1.0i + 1.0j + 1.0k

julia> a = normalize(q)
0.5 + 0.5i + 0.5j + 0.5k

julia> abs(a)
1.0

rotate(v::Vector, q::Quaternion, center=zeros(3))
rotate(v::Vector; axis, angle, center=zeros(3))

Rotate a vector v by a quaternion q around a central point center. The quaternion may be given directly or as an axis and an angle. The point pt is optional and defaults to the origin.

axistoquaternion(axis, angle)

Receive an axis v and angle θ and return the Quaternion corresponding to a rotation of θ around v.

quaterniontomatrix(q::Quaternion)

Return the rotation matrix associated with a Quaternion.

matrixtoquaternion(R)

Given a rotation matrix R, return a unit quaternion q such that rotate(v,q) = R*v for all v.

The matrix R is assumed to be orthogonal but, for efficiency reasons, no check is made to guarantee that.

Since there are, in general, two unit quaternions representing the same rotation matrix, it is not guaranteed that matrixtoquaternion ∘ quaterniontomatrix equals the identity.