Quaternions as Geometric Algebra

Quaternions arise naturally as the even subalgebra of the geometric algebra over three-dimensional Euclidean space. This page derives the conventions used throughout the package from first principles — starting from the geometric product — so that every sign choice, basis assignment, and rotation formula has a clear geometric justification.

The treatment here focuses on real quaternions. The extension to complex quaternions (needed for Lorentz boosts via the spacetime algebra) is covered on a separate page.

The geometric algebra over ℝ³

We start with the standard right-handed Cartesian coordinate system and its unit basis vectors $(𝐱, 𝐲, 𝐳)$. The geometric product of two vectors $𝐯$ and $𝐰$ is defined by

\[𝐯 𝐰 = 𝐯 ⋅ 𝐰 + 𝐯 ∧ 𝐰,\]

where the dot product is the usual scalar inner product and the wedge product is the antisymmetric outer product (the exterior product). The geometric product is linear, associative, and distributive, and satisfies

\[𝐯 𝐯 = 𝐯 ⋅ 𝐯.\]

Two key consequences follow immediately. Parallel vectors commute: $𝐯 𝐰 = 𝐰 𝐯$ when $𝐯 ∥ 𝐰$. Orthogonal vectors anticommute: $𝐯 𝐰 = -𝐰 𝐯$ when $𝐯 ⟂ 𝐰$.

The full algebra over ℝ³ has dimension $2^3 = 8$. A basis is provided by products of the Cartesian basis vectors, grouped by grade (number of vector factors):

\[\begin{array}{ll} \text{grade 0 (scalar):} & \boldsymbol{1}, \\[4pt] \text{grade 1 (vectors):} & 𝐱,\; 𝐲,\; 𝐳, \\[4pt] \text{grade 2 (bivectors):} & 𝐱𝐲,\; 𝐱𝐳,\; 𝐲𝐳, \\[4pt] \text{grade 3 (pseudoscalar):} & 𝐈 = 𝐱𝐲𝐳. \end{array}\]

Each bivector squares to $-1$. For example,

\[𝐱𝐲𝐱𝐲 = -𝐱𝐲𝐲𝐱 = -𝐱(𝐲𝐲)𝐱 = -𝐱𝐱 = -1.\]

The pseudoscalar $𝐈 = 𝐱𝐲𝐳$ also squares to $-1$, so its inverse is $𝐈^{-1} = -𝐈 = 𝐳𝐲𝐱$.

Duality

$𝐈$ has a special property in three dimensions: it commutes with every element of the algebra, so left- and right-multiplication by it are equivalent. This makes it possible to use $𝐈$ (or $𝐈^{-1}$) to map between vectors and bivectors in a way analogous to the Hodge star operator. Explicit computation gives

\[𝐈\,𝐱 = 𝐲𝐳, \qquad 𝐈\,𝐲 = 𝐳𝐱, \qquad 𝐈\,𝐳 = 𝐱𝐲,\]

while multiplication by $𝐈^{-1} = -𝐈$ flips each sign:

\[𝐈^{-1}𝐱 = 𝐳𝐲, \qquad 𝐈^{-1}𝐲 = 𝐱𝐳, \qquad 𝐈^{-1}𝐳 = 𝐲𝐱.\]

We will use this duality to define the quaternion units in the next section, and will see that the choice between $𝐈$ and $𝐈^{-1}$ corresponds precisely to the two sign conventions for quaternion multiplication.

The even subalgebra: Quaternions

Products of an even number of vectors — grades 0 and 2 — form a closed subalgebra, the even subalgebra, spanned by $\{𝟏, 𝐱𝐲, 𝐱𝐳, 𝐲𝐳\}$. This four-dimensional algebra is precisely the quaternions.^[1]

The canonical bivector basis: $𝐢$, $𝐣$, $𝐤$

We define the quaternion units as the duals of the Cartesian basis vectors under $𝐈^{-1}$:

\[𝐢 = 𝐈^{-1}𝐱 = 𝐳𝐲, \qquad 𝐣 = 𝐈^{-1}𝐲 = 𝐱𝐳, \qquad 𝐤 = 𝐈^{-1}𝐳 = 𝐲𝐱.\]

Each of these squares to $-1$ (as expected for a bivector), and one can verify the standard cyclic rules:

\[𝐢^2 = 𝐣^2 = 𝐤^2 = 𝐢𝐣𝐤 = -\boldsymbol{1}, \qquad 𝐢𝐣 = 𝐤, \quad 𝐣𝐤 = 𝐢, \quad 𝐤𝐢 = 𝐣.\]

We will see below that this choice ensures $𝐢$ generates right-handed rotations about $𝐱$, $𝐣$ about $𝐲$, and $𝐤$ about $𝐳$.

Using $𝐈$ instead of $𝐈^{-1}$ would give the alternative assignment $𝐢 = 𝐲𝐳$, $𝐣 = 𝐳𝐱$, $𝐤 = 𝐱𝐲$, which flips the sign of every multiplication: $𝐢𝐣𝐤 = +\boldsymbol{1}$. This is the convention used in much of the robotics and aerospace literature.^[2]

Components and storage

A general quaternion is written

\[𝐐 = w\,\boldsymbol{1} + x\,𝐢 + y\,𝐣 + z\,𝐤,\]

and stored as the tuple (w, x, y, z), matching the component names used throughout the package.

The reverse and the quaternion conjugate

The reverse $\widetilde{𝐐}$ of a geometric-algebra element reverses the order of every vector factor in each basis element. For a grade-0 element the reverse is the identity; for a grade-2 bivector it introduces a sign flip (since swapping two anticommuting vectors costs a minus sign):

\[\widetilde{𝐢} = \widetilde{𝐳𝐲} = 𝐲𝐳 = -𝐢, \quad \widetilde{𝐣} = -𝐣, \quad \widetilde{𝐤} = -𝐤.\]

For a general quaternion this gives

\[\widetilde{𝐐} = w - x\,𝐢 - y\,𝐣 - z\,𝐤,\]

which is exactly the quaternion conjugate conj(Q) in the code.

Two important consequences follow. First,

\[𝐐\,\widetilde{𝐐} = w^2 + x^2 + y^2 + z^2\]

is a pure scalar — a non-negative real number for real quaternions. In analogy with abs2 for complex numbers, this product is abs2(Q) in the code. Second, every nonzero quaternion has an inverse

\[𝐐^{-1} = \frac{\widetilde{𝐐}}{𝐐\,\widetilde{𝐐}} = \frac{\texttt{conj}(Q)}{\texttt{abs2}(Q)}.\]

Real vs. complex quaternions

For real quaternions, $𝐐\,\widetilde{𝐐}$ is a positive real scalar. For complex quaternions, $𝐐\,\widetilde{𝐐}$ is still a scalar element of the algebra (no $𝐢$, $𝐣$, or $𝐤$ component), but it is complex-valued — it is the spinor norm $\sum_i z_i^2$ rather than the Euclidean norm $\sum_i |z_i|^2$. The distinction matters for Lorentz boosts; it is discussed on the spacetime algebra page.

Rotors and the double cover of SO(3)

A rotor is a unit quaternion — a quaternion satisfying $𝐑\,\widetilde{𝐑} = \boldsymbol{1}$. Any unit quaternion can be written

\[𝐑 = \exp\!\left(\frac{\rho}{2}\,\hat{𝔯}\right) = \cos\frac{\rho}{2} + \hat{𝔯}\,\sin\frac{\rho}{2},\]

where $\rho$ is a real angle and $\hat{𝔯} = \hat{r}_x 𝐢 + \hat{r}_y 𝐣 + \hat{r}_z 𝐤$ is a unit pure-vector quaternion.

The group of unit quaternions is $\mathrm{Spin}(3) \cong \mathrm{SU}(2)$, topologically the 3-sphere $\mathbb{S}^3$. It is a double cover of the rotation group $\mathrm{SO}(3)$: both $𝐑$ and $-𝐑$ represent the same rotation (as we will see below), but they are distinct quaternions representing distinct spinors.

The vector–quaternion isomorphism and rotations

The isomorphism

The duality relations $𝐢 = 𝐈^{-1}𝐱$, etc., provide a natural identification between ℝ³ and the space of pure-vector quaternions:

\[𝐱 \leftrightarrow 𝐢, \qquad 𝐲 \leftrightarrow 𝐣, \qquad 𝐳 \leftrightarrow 𝐤.\]

This is more than a vector-space isomorphism. For any two pure quaternions $𝐩 = p_x 𝐢 + p_y 𝐣 + p_z 𝐤$ and $𝐪 = q_x 𝐢 + q_y 𝐣 + q_z 𝐤$, direct expansion of the geometric product gives

\[𝐩\,𝐪 = -(𝐩 \cdot 𝐪) + (𝐩 \times 𝐪),\]

where the dot product contributes a scalar and the cross product contributes a pure quaternion via the same identification. This is the exact analogue of the geometric product for vectors, $𝐯𝐰 = 𝐯 \cdot 𝐰 + 𝐯 \wedge 𝐰$, with Hodge duality mapping $𝐯 \wedge 𝐰 \leftrightarrow 𝐯 \times 𝐰$. In particular, the antisymmetric part gives

\[\frac{𝐩\,𝐪 - 𝐪\,𝐩}{2} = 𝐩 \times 𝐪,\]

which is precisely the Lie bracket of the cross-product algebra on ℝ³. The map is therefore an isomorphism of Lie algebras.

Under this identification, a vector $𝐯 = v_x 𝐱 + v_y 𝐲 + v_z 𝐳$ is represented by the pure quaternion $v_x 𝐢 + v_y 𝐣 + v_z 𝐤$, and we freely pass between the two descriptions throughout the package.

The rotation formula

A rotor $𝐑$ acts on a vector $𝐯$ (written as a pure quaternion) by the sandwich product

\[𝐯' = 𝐑\, 𝐯\, 𝐑^{-1} = 𝐑\, 𝐯\, \widetilde{𝐑}.\]

To see that this is a right-handed rotation through angle $\rho$ about the axis $\hat{𝔯}$, decompose $𝐯 = 𝐯_\parallel + 𝐯_\perp$ into parts parallel and perpendicular to $\hat{𝔯}$.

$𝐯_\parallel$ commutes with $\hat{𝔯}$ (parallel vectors commute), so it commutes with $𝐑$ and passes through unchanged.
$𝐯_\perp$ anticommutes with $\hat{𝔯}$ (orthogonal vectors anticommute), so the two factors of $𝐑$ act non-trivially. Expanding:

\[\begin{aligned} 𝐑\, 𝐯_\perp\, 𝐑^{-1} &= \left(\cos\tfrac{\rho}{2} + \sin\tfrac{\rho}{2}\,\hat{𝔯}\right) 𝐯_\perp \left(\cos\tfrac{\rho}{2} - \sin\tfrac{\rho}{2}\,\hat{𝔯}\right) \\ &= \cos^2\!\tfrac{\rho}{2}\; 𝐯_\perp + \sin\tfrac{\rho}{2}\cos\tfrac{\rho}{2}\,[\hat{𝔯}, 𝐯_\perp] - \sin^2\!\tfrac{\rho}{2}\;\hat{𝔯}\, 𝐯_\perp\, \hat{𝔯} \\ &= \cos\rho\; 𝐯_\perp + \sin\rho\; \hat{𝔯} \times 𝐯_\perp, \end{aligned}\]

which is precisely the right-handed rotation of $𝐯_\perp$ through angle $\rho$ about $\hat{𝔯}$. Putting the two parts together:

\[𝐑\, 𝐯\, 𝐑^{-1} = 𝐯_\parallel + \cos\rho\; 𝐯_\perp + \sin\rho\; \hat{𝔯} \times 𝐯_\perp.\]

The presence of two factors of $𝐑$ in the sandwich explains two things. First, the rotation angle is twice the quaternion half-angle: each factor of $𝐑 = \cos(\rho/2) + \ldots$ contributes a half-angle, and together they give the full angle $\rho$. Second, negating $𝐑 \to -𝐑$ leaves the sandwich unchanged, confirming the double cover.

Rotations about coordinate axes

As a concrete check, the rotor for a 90° rotation about $𝐳$ is

\[𝐑 = \exp\!\left(\frac{\pi}{4} 𝐤\right) = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}\, 𝐤.\]

Applying it to $𝐢$ (corresponding to $𝐱$):

\[𝐑\, 𝐢\, 𝐑^{-1} = \left(\tfrac{1}{\sqrt{2}} + \tfrac{1}{\sqrt{2}} 𝐤\right) 𝐢 \left(\tfrac{1}{\sqrt{2}} - \tfrac{1}{\sqrt{2}} 𝐤\right) = 𝐣,\]

confirming a right-handed rotation $𝐱 \to 𝐲$ about $𝐳$.

Efficient computation

In the package, the function-call syntax R(v) implements the sandwich $𝐑\, 𝐯\, \widetilde{𝐑}$ efficiently with a dedicated formula — roughly twice as fast as the three-multiplication sequence R * v * conj(R):

Q * v * conj(Q) ≈ Q(v)   # both correct; Q(v) is ~2× faster

This applies to both Quaternion and Rotor objects acting on a QuatVec.