Читать книгу Algebra and Applications 1 - Abdenacer Makhlouf - Страница 30

Given a norm 1 quaternion q, there is an angle α ∈ [0, π] and a norm 1 imaginary quaternion u such that q = (cos α)1 + (sin α)u.

Consider the linear map:

Complete u to an orthonormal basis {u, v, u × v}. A simple computation gives:

Thus, the coordinate matrix of φq relative to the basis {u, v, u × v} is

In other words, φq is a rotation around the semi-axis ℝ⁺u of angle 2α, and hence the map

is a surjective (Lie) group homomorphism with ker φ = {±1}. We thus obtain the isomorphism

Actually, the group S³ is the universal cover of SO₃(ℝ).

Therefore, we get that rotations can be identified with conjugation by norm 1 quaternions modulo ±1. The outcome is that it is quite easy now to compose rotations in three-dimensional Euclidean space, as it is enough to multiply norm 1 quaternions: φp φq = φpq. From this, one can very easily deduce the 1840 formulas by Olinde Rodrigues (Rodrigues 1840) for the composition of rotations.

But there is more about rotations and quaternions.

For any p ∈ ℍ with n(p) = 1, the left (respectively, right) multiplication Lp (respectively, Rp) by p is an isometry, due to the multiplicativity of the norm. Using [2.2], it follows that the characteristic polynomial of Lp and Rp is x² − tr(p)x +1)² and, in particular, the determinant of the multiplication by p is 1, so both Lp and Rp are rotations.

Now, if ψ is a rotation in ℝ⁴ ≃ ℍ, a = ψ(1) is a norm 1 quaternion, and

so the composition is actually a rotation in ℝ³ ≃ ℍ₀. Hence, there is a norm 1 quaternion q ∈ ℍ such that

for any x ∈ ℍ. That is, for any x ∈ ℍ,

It follows that the map

is a surjective (Lie) group homomorphism with ker Ψ = {±(1, 1)}. We thus obtain the isomorphism

and from here we get the isomorphism SO₃ (ℝ) × SO₃ (ℝ) ≃ PSO₄ (ℝ).

Again, this means that it is easy to compose rotations in four-dimensional space, as it reduces to multiplying pairs of norm 1 quaternions: ψp_{1, q1} ψp_{2, q2} = ψp_{1p2, q1q2}.