Quaternion-Based Geometric Interpretation

• Quaternion-based geometric interpretation is a mathematical framework that uses quaternions to represent rotations and symmetries in multidimensional spaces, linking algebra with topology and spinor theory.
• It employs a quaternionic representation for axis-angle mapping and utilizes the 3-sphere as a double cover of SO(3), showcasing the unique 720° rotational identity property.
• This approach underpins advanced applications in quantum mechanics and computer graphics by offering efficient interpolation and robust representations of spatial transformations.

A quaternion-based geometric interpretation establishes a unified mathematical and conceptual framework for understanding multidimensional rotations, symmetries, and mappings by leveraging the algebraic structure of quaternions and their deep connections to geometry, topology, and spinor theory. Quaternions—the four-dimensional associative real division algebra generated by units {1, i, j, k} with Hamilton’s multiplication rules—naturally encode and generalize essential geometric operations in ℝ³ and ℝ⁴, including rotations, interpolations, and more exotic constructions such as the Hopf fibration and gauge connections.

1. Quaternions and the 3-Sphere: Double Cover of SO(3)

A quaternion $q = q_0 + q_1 i + q_2 j + q_3 k$ decomposes into a real scalar part and a three-vector imaginary part. The set of unit quaternions $|q| = \sqrt{q_0^2 + q_1^2 + q_2^2 + q_3^2} = 1$ forms a 3-sphere $S^3 \subset \mathbb{R}^4$. Every element of $SO(3)$ corresponds to exactly two antipodal points, $\pm q$, on $S^3$, making $S^3$ a double cover of $SO(3)$ (Parwana et al., 2017). This reflects the topological fact that a 720° ($4\pi$) rotation in 3D space is homotopic to the identity in $S^3$ but not in $SO(3)$, and underpins the fundamental $SU(2) \to SO(3)$ correspondence relevant for quantum spin and topology (Krishnaswami et al., 2016, Palais, 2010).

2. Quaternionic Representation of Rotations and Axis-Angle Mapping

By Euler’s rotation theorem, any rotation can be expressed as a rotation by an angle $\theta$ about a unit axis $\hat{n}$. The corresponding unit quaternion is

$$q = [q_w, \mathbf{q}_v] = [\cos(\theta/2), \hat{n} \sin(\theta/2)].$$

Conversely, for $q = [q_w, \mathbf{q}_v]$ with $|q| = 1$: $$\theta = 2 \arccos(q_w),\quad \hat{n} = \mathbf{q}_v/\sin(\theta/2) \ ( \sin(\theta/2) \neq 0)$$ (Parwana et al., 2017, Hanson, 2018).

The geometric action on a vector $\mathbf{x} \in \mathbb{R}^3$ is given by quaternion conjugation: $$\mathbf{x}' = q \otimes [0,\mathbf{x}] \otimes q^*,$$ inducing a proper rotation $R(q) \in SO(