Complex Geometry: Planar Rotors and Interval Arcs

2.1 Complex Numbers as Even Subalgebra

In \(\mathcal{G}(\mathbb{R}^2)\), the basis vectors \(\mathbf{e}_1\) and \(\mathbf{e}_2\) satisfy \(\mathbf{e}_1^2 = \mathbf{e}_2^2 = 1\) and \(\mathbf{e}_1\mathbf{e}_2 = -\mathbf{e}_2\mathbf{e}_1\). Define the unit bivector:

\[ \mathbf{I} = \mathbf{e}_1 \wedge \mathbf{e}_2 = \mathbf{e}_1 \mathbf{e}_2 \]

Computing its square: \(\mathbf{I}^2 = (\mathbf{e}_1\mathbf{e}_2)(\mathbf{e}_1\mathbf{e}_2) = \mathbf{e}_1(\mathbf{e}_2\mathbf{e}_1)\mathbf{e}_2 = -\mathbf{e}_1^2\mathbf{e}_2^2 = -1\). The bivector \(\mathbf{I}\) squares to \(-1\) — not as a postulate, but as a theorem of the geometric product.

Theorem 2.1 — Complex numbers as even subalgebra

The even-graded subalgebra \(\mathcal{G}^+(\mathbb{R}^2) = \{a + b\mathbf{I} \mid a,b \in \mathbb{R}\}\) is isomorphic to the complex numbers \(\mathbb{C}\), with \(\mathbf{I} \leftrightarrow i\). This is a structural identity, not a convention.^{[1, §3.2]}

The significance is epistemological as much as algebraic: complex numbers are not an extension of the real numbers requiring a leap of abstraction — they are the natural algebra of the oriented plane, already present in the structure of 2D space. Lounesto^[2] develops this theme in detail for all Clifford algebras.

2.2 Rotors: The Exponential Map

A rotor is a unit-magnitude even multivector that implements rotation via the sandwich product. In 2D, a rotor encoding a rotation by angle \(\theta\) is:

\[ R = e^{-\mathbf{I}\theta/2} = \cos(\theta/2) - \mathbf{I}\sin(\theta/2) \]

and the rotation of a vector \(\mathbf{v}\) is:

\[ \mathbf{v}' = R\,\mathbf{v}\,R^\dagger = R\,\mathbf{v}\,\tilde{R} \]

where \(\tilde{R}\) is the reverse of \(R\). Note that the half-angle parameterisation is essential for this to work in all dimensions; in 2D the formula simplifies and the full angle can be used directly in the exponential.

Fig. 2.1. Left: the interval rotor \(R([\theta_1, \theta_2])\) sweeps an arc-sector on the unit circle. Right: when applied to a vector \(\mathbf{v}\), the interval rotor maps \(\mathbf{v}\) to a sector — the geometric representation of phase uncertainty. Compare Hestenes [1, Ch. 5].

2.3 Interval Rotors and Phase Uncertainty

A classical complex interval \([z_1, z_2]\) is typically interpreted as a rectangular region in \(\mathbb{C}\). In GA, we have a richer choice: we can parameterise uncertainty by magnitude and phase separately. An interval rotor is:

\[ R([\theta_1, \theta_2]) = \{ e^{\mathbf{I}\theta} \mid \theta \in [\theta_1, \theta_2] \} \]

This represents an arc on the unit circle — all possible pure rotations within the angular range. Combined with magnitude uncertainty \([r_1, r_2]\), the full interval complex number becomes an annular sector:

\[ [z] = \{ r e^{\mathbf{I}\theta} \mid r \in [r_1, r_2],\ \theta \in [\theta_1, \theta_2] \} \]

Definition 2.2 — Annular sector interval

The polar interval complex number \([r_1,r_2]\cdot e^{\mathbf{I}[\theta_1,\theta_2]}\) is the annular sector with inner radius \(r_1\), outer radius \(r_2\), initial angle \(\theta_1\), and final angle \(\theta_2\). It is the natural representation of a complex number with independently bounded magnitude and phase.^[3]

2.4 Application: Uncertain Phase in Differential Equations

Consider a linear ODE \(\dot{z} = \lambda z\) where \(\lambda = \sigma + \mathbf{I}\omega\) is a complex eigenvalue with uncertain imaginary part: \(\omega \in [\omega_1, \omega_2]\). The solution is:

\[ z(t) = z_0 \, e^{\sigma t} \cdot e^{\mathbf{I}\omega t} \]

Since \(\omega\) is uncertain, the "phase" of the solution \(e^{\mathbf{I}\omega t}\) lies in the arc \(e^{\mathbf{I}[\omega_1 t, \omega_2 t]}\). Over time, this arc widens: the longer we evolve the system, the larger the angular uncertainty in the solution's phase.

// Solution with uncertain frequency ω ∈ [ω₁, ω₂] z(t) = z₀ · e^(σt) · e^(I·[ω₁t, ω₂t]) // Angular spread of phase uncertainty at time t Δθ(t) = (ω₂ - ω₁) · t // grows linearly // For a stable system (σ < 0), the magnitude contracts // but the phase sector can still widen — the solution cloud // spirals inward while fanning out angularly.

Teaching insight. Students often assume that a stable system (negative real part of eigenvalue) is "safe" in every sense. But phase uncertainty grows even when the amplitude contracts. The annular sector representation makes this immediately visible: the sector narrows radially while widening angularly.

Fig. 2.2. Evolution of the interval complex number under \(\dot{z} = (\sigma + \mathbf{I}\omega)z\) with \(\sigma < 0\) (stable) and \(\omega \in [\omega_1, \omega_2]\). The annular sector contracts radially but fans out angularly as the phase uncertainty \((\omega_2-\omega_1)t\) increases linearly with time.

2.5 The Blade Map: from \(\mathbb{C}\) to \(\mathcal{G}^+(\mathbb{R}^2)\)

Classical complex number

\(z = a + bi\)
where \(i\) is "imaginary," defined by \(i^2 = -1\) as an axiom.

Geometric algebra element

\(Z = a + b\mathbf{I}\)
where \(\mathbf{I} = \mathbf{e}_1\mathbf{e}_2\) is a bivector, and \(\mathbf{I}^2 = -1\) is a theorem.

This mapping is more than cosmetic. The bivector interpretation of \(i\) reveals why multiplication by \(i\) rotates: because multiplying by the unit bivector \(\mathbf{I}\) geometrically implements a 90° rotation in the \(\mathbf{e}_1\mathbf{e}_2\) plane. Hestenes^[1] shows this extends to quaternions (as the even subalgebra of \(\mathcal{G}(\mathbb{R}^3)\)) and spinors more generally.

References

Hestenes, D. (1999). New Foundations for Classical Mechanics (2nd ed.). Kluwer Academic. Chapter 5.
Lounesto, P. (2001). Clifford Algebras and Spinors (2nd ed.). Cambridge University Press.
Rokne, J. G. (2001). Interval arithmetic and interval analysis: An introduction. In Granular Computing. Physica-Verlag.
Macdonald, A. (2011). Linear and Geometric Algebra. CreateSpace. §10.