Ray-Centered Rotation Parameterization

• Ray-centered rotation parameterization is a minimal, algebraic SO(3) representation that avoids singularities and enables closed-form forward and inverse computation.
• It facilitates efficient variational integrators with robust numerical updates, eliminating the need for quaternion normalization in rigid body dynamics.
• In neutrino physics, it offers a compact one-axis description of the PMNS matrix, enforcing CP conservation and yielding predictive mixing parameters.

The ray-centered rotation parameterization—also known as the rescaled Rodrigues or “exponential local coordinate” rotation representation—encodes elements of SO(3) in terms of a scaled axis–angle vector. Originally developed to enable explicit, second-order variational time integrators for rigid body dynamics, it is distinguished by its rational functional form, algebraic composition properties, and avoidance of typical singularities associated with Euler angles and quaternion normalization. This parameterization has further found utility in the compact description of particle mixing matrices in flavor physics, by enabling a single-axis rotation representation of the standard PMNS matrix. Its technical distinctiveness stems from facilitating closed-form computation for both forward and inverse rotation maps, as well as highly stable incremental updates in numerical simulation and physical modeling contexts (Baker et al., 2021, Duda et al., 29 Sep 2025).

1. Mathematical Formulation

Given a finite rotation about a unit axis $u \in S^2$ by an angle $\theta$, the ray-centered (rescaled Rodrigues) parameter $\phi \in \mathbb{R}^3$ is defined as: $$\phi = s(\theta)\,u, \qquad s(\theta) = \frac{\tan(\theta/2)}{\theta/2}$$ Equivalently,

$$\phi = \frac{\tan(\|\theta\|/2)}{\|\theta\|/2} \theta$$

The associated rotation matrix $R(\phi) \in SO(3)$ results from the closed-form expression: $$R(\phi) = I + \frac{4}{4+\|\phi\|^2}\left[ S(\phi) + \frac{1}{2}S(\phi)^2 \right]$$ where $S(\phi)$ denotes the $3\times3$ skew-symmetric cross-product matrix. This formula is purely algebraic in $\phi$, eliminating trigonometric computations. The inverse map, recovering axis and angle from $\theta$0, utilizes: $\theta$1 Yielding

$\theta$2

(Baker et al., 2021).

2. Differential Properties and Jacobians

The differential $\theta$3 required for sensitivity analysis and optimization is computed by differentiating the algebraic closed-form $\theta$4. The resulting expressions, which combine the derivative of the rational scalar prefactor and bilinear products of $\theta$5, allow explicit construction of the left and right Jacobians $\theta$6 for $\theta$7. Although explicit tabulation is not provided, these forms are directly suitable for numerical and analytical differentiation in algorithms requiring efficient mapping between tangent vectors and rotations (Baker et al., 2021).

3. Algebraic Closure and Composition

The ray-centered parameterization supports algebraic composition of rotations directly in parameter space. If two rotations are encoded by $\theta$8, the composition $\theta$9 is given by: $\phi \in \mathbb{R}^3$0 This preserves group closure and ensures that composed parameters correspond to a valid $\phi \in \mathbb{R}^3$1 element without projection or normalization constraints. The method remains non-singular for $\phi \in \mathbb{R}^3$2 ($\phi \in \mathbb{R}^3$3), providing robust behavior for small to moderate angular increments, which is critical in variational integration and physical simulation (Baker et al., 2021).

4. Numerical and Computational Attributes

This parameterization offers several computational benefits:

• The absence of trigonometric functions in $\phi \in \mathbb{R}^3$4 leads to improved performance and numerical stability.
• Updates to rotation state through algebraic formulas avoid drift and normalization overhead seen in quaternion integration.
• The representation avoids Euler angle–like singularities (gimbal lock) and stays globally well-posed except as $\phi \in \mathbb{R}^3$5.
• Stability under incremental updates is maintained provided $\phi \in \mathbb{R}^3$6; the explicit variational integrator framework imposes such bounds for stepwise integration.
• Facilitates second-order time integrators that conserve discrete momentum maps exactly and maintain near-conservation of total energy over exponentially long simulation durations.

These properties directly support the construction of explicit variational integrators for SO(3) and SE(3) systems, with demonstrated applications including the rigid body pendulum and particle-binder collision dynamics (Baker et al., 2021).

5. Applications in Neutrino Mixing and Group Structure

An analogous ray-centered (single-axis rotation) parameterization is applied in particle physics to represent the neutrino mixing (PMNS) matrix as a single $\phi \in \mathbb{R}^3$7 rotation: $\phi \in \mathbb{R}^3$8 or, equivalently, $\phi \in \mathbb{R}^3$9 with $$\phi = s(\theta)\,u, \qquad s(\theta) = \frac{\tan(\theta/2)}{\theta/2}$$0, where $$\phi = s(\theta)\,u, \qquad s(\theta) = \frac{\tan(\theta/2)}{\theta/2}$$1 are the SO(3) generators. This compactly encodes the full mixing in terms of an axis $$\phi = s(\theta)\,u, \qquad s(\theta) = \frac{\tan(\theta/2)}{\theta/2}$$2 and angle $$\phi = s(\theta)\,u, \qquad s(\theta) = \frac{\tan(\theta/2)}{\theta/2}$$3, corresponding numerically to projections $$\phi = s(\theta)\,u, \qquad s(\theta) = \frac{\tan(\theta/2)}{\theta/2}$$4, $$\phi = s(\theta)\,u, \qquad s(\theta) = \frac{\tan(\theta/2)}{\theta/2}$$5, $$\phi = s(\theta)\,u, \qquad s(\theta) = \frac{\tan(\theta/2)}{\theta/2}$$6, with $$\phi = s(\theta)\,u, \qquad s(\theta) = \frac{\tan(\theta/2)}{\theta/2}$$7 and $$\phi = s(\theta)\,u, \qquad s(\theta) = \frac{\tan(\theta/2)}{\theta/2}$$8 (Duda et al., 29 Sep 2025).

The enforced reality of the $$\phi = s(\theta)\,u, \qquad s(\theta) = \frac{\tan(\theta/2)}{\theta/2}$$9 form fixes the CP phase $$\phi = \frac{\tan(\|\theta\|/2)}{\|\theta\|/2} \theta$$0 (or equivalently, $$\phi = \frac{\tan(\|\theta\|/2)}{\|\theta\|/2} \theta$$1). The decomposition projects onto standard solar, reactor, and atmospheric mixing angles. The allowed values for $$\phi = \frac{\tan(\|\theta\|/2)}{\|\theta\|/2} \theta$$2 and $$\phi = \frac{\tan(\|\theta\|/2)}{\|\theta\|/2} \theta$$3 span the entire region permitted by current oscillation data for $$\phi = \frac{\tan(\|\theta\|/2)}{\|\theta\|/2} \theta$$4. This framework produces sharp predictions for phenomena such as tritium $$\phi = \frac{\tan(\|\theta\|/2)}{\|\theta\|/2} \theta$$5-decay and neutrinoless double-$$\phi = \frac{\tan(\|\theta\|/2)}{\|\theta\|/2} \theta$$6 decay, with the latter’s effective mass $$\phi = \frac{\tan(\|\theta\|/2)}{\|\theta\|/2} \theta$$7 forced near the maximal edge of the normal ordering "funnel" (Duda et al., 29 Sep 2025).

6. Comparative and Practical Considerations

The ray-centered parameterization contrasts with Euler angles and quaternion-based systems:

• It encodes $$\phi = \frac{\tan(\|\theta\|/2)}{\|\theta\|/2} \theta$$8 elements as a minimal, $$\phi = \frac{\tan(\|\theta\|/2)}{\|\theta\|/2} \theta$$9-vector in $R(\phi) \in SO(3)$0 with direct algebraic closure and global non-singularity for $R(\phi) \in SO(3)$1.
• All updates remain in $R(\phi) \in SO(3)$2 by construction, preventing the need for ad hoc normalization.
• The algebraic, trigonometric-free mapping accelerates computation in integrator routines.
• In neutrino physics, the one-parameter $R(\phi) \in SO(3)$3 description is both more restrictive and predictive: it implies exact CP conservation and symmetric appearance probabilities in oscillation experiments (neutrino$R(\phi) \in SO(3)$4antineutrino symmetry). Any significant deviation of $R(\phi) \in SO(3)$5 from $R(\phi) \in SO(3)$6 would decisively falsify the single-rotation hypothesis (Duda et al., 29 Sep 2025).

7. Significance and Outlook

The ray-centered rotation parameterization provides a technically robust and algebraically concise representation of three-dimensional rotations. In computational mechanics, it serves as the backbone for explicit variational integrators with provable conservation properties and efficient numerical realization. In the context of group structure and neutrino phenomenology, it offers a compact and physically distinctive alternative to three-angle (Euler) parameterizations, with pronounced predictive implications for current and next-generation experimental tests. The parameterization’s clean closure, computational efficiency, and global stability continue to shape developments in simulation, optimization, and the representation theory of $R(\phi) \in SO(3)$7 (Baker et al., 2021, Duda et al., 29 Sep 2025).