Cayley Parameterization of Rotations

• Cayley parameterization of rotations is a rational method that converts skew-symmetric matrices into SO(n) rotations without using transcendental functions.
• It offers explicit, closed-form formulas in low dimensions (e.g., SO(2) and SO(3)), which simplify pose estimation and kinematic computations.
• The algebraic nature of the method supports applications in robotics, computer vision, machine learning, and quantum mechanics by enabling efficient derivative evaluations and optimization.

Cayley parameterization of rotations is a classical algebraic construction that provides a rational and effective means for representing elements of the special orthogonal group SO(n), the group of proper rotations in n-dimensional Euclidean space, via skew-symmetric (Lie algebra) generators. This parameterization, initiated by Arthur Cayley in the 19th century, is extensively developed in modern applications ranging from robotics and computer vision to numerical analysis, machine learning, probability, and quantum mechanics. The construction exploits simple matrix operations and avoids transcendental functions, yielding computational and analytic advantages in many contexts.

1. Definition and Fundamental Properties

Given a real n×n skew-symmetric matrix $A \in \mathfrak{so}(n)$ (i.e., $A^\top = -A$), the classical Cayley transform is defined as

$R = \mathrm{Cay}(A) := (I - A)(I + A)^{-1}$

provided that $(I + A)$ is invertible. The resulting matrix $R$ is guaranteed to be orthogonal ($R^\top R = I$) with determinant $+1$ ($\det R = 1$), hence a rotation, i.e., $R \in \mathrm{SO}(n)$ (Pauli et al., 2023, Jauch et al., 2018).

The map $\mathrm{Cay}: \mathfrak{so}(n) \to \mathrm{SO}(n)$ is bijective onto the subset of $A^\top = -A$0 consisting of those rotations for which $A^\top = -A$1 is not an eigenvalue (i.e., those not corresponding to a rotation by angle $A^\top = -A$2 in any invariant plane). The inverse Cayley transform is

$A^\top = -A$3

again defined when $A^\top = -A$4 is invertible, i.e., for $A^\top = -A$5 with no $A^\top = -A$6 eigenvalues (Macías-Virgós et al., 2016, Mahmoudi, 2016).

The transform is real-analytic, rational in the entries of $A^\top = -A$7, and infinitely differentiable on its domain (Pauli et al., 2023).

2. Domain, Surjectivity, and Topological Aspects

Domain and Range

• Domain: All $A^\top = -A$8 such that $A^\top = -A$9, equivalently, $R = \mathrm{Cay}(A) := (I - A)(I + A)^{-1}$0.
• Range: All rotations $R = \mathrm{Cay}(A) := (I - A)(I + A)^{-1}$1 with $R = \mathrm{Cay}(A) := (I - A)(I + A)^{-1}$2. Rotations by precisely angle $R = \mathrm{Cay}(A) := (I - A)(I + A)^{-1}$3 (in any invariant two-plane) are excluded (Pauli et al., 2023, Jauch et al., 2018, Mahmoudi, 2016).

Local Behavior and Topology

Cayley transform is a real-analytic diffeomorphism between the domain in $R = \mathrm{Cay}(A) := (I - A)(I + A)^{-1}$4 and the open subset of SO(n) avoiding $R = \mathrm{Cay}(A) := (I - A)(I + A)^{-1}$5 eigenvalues. This exclusion is a measure-zero subset from the perspective of the Haar measure and the smooth structure (Jauch et al., 2018).

3. Explicit Formulas in Low Dimensions

SO(2)

For $R = \mathrm{Cay}(A) := (I - A)(I + A)^{-1}$6, the Cayley transform yields

$R = \mathrm{Cay}(A) := (I - A)(I + A)^{-1}$7

where $R = \mathrm{Cay}(A) := (I - A)(I + A)^{-1}$8. As $R = \mathrm{Cay}(A) := (I - A)(I + A)^{-1}$9, $(I + A)$0, and the map becomes singular (Pauli et al., 2023, Mahmoudi, 2016).

SO(3)

A general $(I + A)$1 for $(I + A)$2, yields

$(I + A)$3

where $(I + A)$4 is the standard cross-product matrix. The resulting rotation is about axis $(I + A)$5 with angle $(I + A)$6. The explicit formula is

$(I + A)$7

The denominator enforces the exclusion of rotations with $(I + A)$8 ($(I + A)$9) (Pauli et al., 2023, Mahmoudi, 2016, Jauch et al., 2018).

Component Formula Table (n=3)

Entry	Formula (with $R$0)
$R$1	$R$2
$R$3	$R$4
$R$5	$R$6
$R$7	$R$8
$R$9	$R^\top R = I$0
$R^\top R = I$1	$R^\top R = I$2
$R^\top R = I$3	$R^\top R = I$4
$R^\top R = I$5	$R^\top R = I$6
$R^\top R = I$7	$R^\top R = I$8

This rational structure is a hallmark of the Cayley parameterization (Martyushev, 2011, Jauch et al., 2018, Mahmoudi, 2016).

4. Analytic Properties, Jacobians, and Comparison to Other Parameterizations

Analytic Features and Jacobian

• The Cayley transform is polynomial or rational in all entries, facilitating efficient evaluation and differentiation.
• The Jacobian determinant for $R^\top R = I$9 is given by $+1$0, directly encoding the volume change under the map from Lie algebra to rotation group (Jauch et al., 2018).

Comparison Table

Parameterization	Domain	Surjectivity	Regularity	Notable Features
Cayley	$+1$1, $+1$2	Not onto full SO(n) (misses angle $+1$3)	Rational, analytic	Simple algebraic form, no transcendentals; excludes measure-zero subset
Exponential	$+1$4	Onto SO(n) ($+1$5); onto component of I ($+1$6)	Analytic, transcendental	Requires trigonometric functions; global but multi-valued (Pauli et al., 2023, Kruglov et al., 2017)
Quaternions (n=3)	$+1$7	Double cover	Rational entries, with unit-norm constraint	Smooth, covers all SO(3); $+1$8 redundancy (Pauli et al., 2023, Kruglov et al., 2017)
Euler Angles	$+1$9	Local charts, possible singularities	Transcendental	Minimal, but can suffer “gimbal lock” (Pauli et al., 2023)

5. Algebraic and Computational Advantages

• The Cayley map converts an unconstrained Euclidean vector space (of skew-symmetric matrices/vectors) into rotations, enabling parametric optimization on SO(n) without explicit orthogonality constraints (Pauli et al., 2023, Jauch et al., 2018, Barfoot et al., 2021).
• The rational formula enables efficient computation of derivatives and Jacobians, particularly advantageous for optimization and geometric integration (Pauli et al., 2023, Jauch et al., 2018, Kortryk, 2015, Barfoot et al., 2021).
• For $\det R = 1$0, the composition of two Cayley-parameterized rotations admits a closed-form algebraic expression: for $\det R = 1$1,

$\det R = 1$2

so that $\det R = 1$3, facilitating direct computations (Valdenebro, 2016).

In high-dimensional settings (large $\det R = 1$4), the cost of inverting $\det R = 1$5 may become a computational bottleneck; however, for structured $\det R = 1$6 (e.g., sparse or low-rank), this can often be mitigated (Pauli et al., 2023, Jauch et al., 2018).

6. Extensions and Applications

Machine Learning, Probability, and Optimization

• Parameterizations based on Cayley transforms are employed to design orthogonal (e.g., rotation-invariant) layers in convolutional neural networks, enforcing Lipschitz constraints and robustness by ensuring spectral norm control (Pauli et al., 2023).
• In probabilistic modeling and simulation, the Cayley coordinates allow unconstrained Markov chain Monte Carlo on SO(n), as densities can be reparametrized into Euclidean space. Change-of-variable formulas and Haar measure transformations are given in closed form (Jauch et al., 2018).
• The Cayley transform defines local diffeomorphisms on Stiefel and Grassmann manifolds, making it central to optimization algorithms with orthogonality constraints (Macías-Virgós et al., 2016, Jauch et al., 2018).
• In computer vision, explicit polynomial expressions based on Cayley parameters facilitate direct, rational solutions to the relative pose problem (e.g., five-point algorithm for stereo vision) without enforcing cubic matrix constraints (Martyushev, 2011).

Quantum Mechanics and High-Spin Systems

• The Cayley transform serves as a rational alternative to exponential map for representing spin-j rotation matrices in SU(2): $\det R = 1$7, $\det R = 1$8, where $\det R = 1$9 is the angular momentum operator. This yields explicit polynomial expressions for arbitrary spin, with the structure of coefficients governed by central factorial numbers (Kortryk, 2015, Curtright, 2015).
• Rational (non-transcendental) expression is favorable for symbolic manipulation and numerical methods, especially when compared to the Curtright–Fairlie–Zachos (CFZ) polynomials derived from the exponential form.

7. Limitations and Relationship to Alternative Representations

• The exclusion of $R \in \mathrm{SO}(n)$0-rotations (or, more generally, elements of SO(n) with $R \in \mathrm{SO}(n)$1 eigenvalues) is inherent: as the parameter $R \in \mathrm{SO}(n)$2 or $R \in \mathrm{SO}(n)$3 diverges, the associated rotation angle approaches $R \in \mathrm{SO}(n)$4, and the Cayley map becomes singular (Pauli et al., 2023, Valdenebro, 2016, Barfoot et al., 2021).
• Although the Cayley map is not globally surjective, two overlapping Cayley charts suffice to cover all of SO(n); this is analogous to the use of affine patches on real projective space or quaternionic atlases for SO(3), eliminating coordinate singularities present in single-patch Cayley (Goldstein, 2010, Jauch et al., 2018).
• The Cayley parameterization admits no direct link to quaternions or slerp interpolation within the three-parameter form; conversion to and from quaternionic or Euler angle descriptions typically requires additional computation (Kruglov et al., 2017, Valdenebro, 2016).
• In unitary and complex settings (SU(2), Cayley–Klein parameters), the Cayley framework provides rational SU(2) rotation formulas but must be used with care regarding global topology and periodicity (Kortryk, 2015).

• "Lipschitz-bounded 1D convolutional neural networks using the Cayley transform and the controllability Gramian" (Pauli et al., 2023)
• "An Algorithmic Solution to the Five-Point Pose Problem Based on the Cayley Representation of Rotations" (Martyushev, 2011)
• "Random orthogonal matrices and the Cayley transform" (Jauch et al., 2018)
• "Cayley Transform on Stiefel manifolds" (Macías-Virgós et al., 2016)
• "Visualizing Rotations and Composition of Rotations with Rodrigues' Vector" (Valdenebro, 2016)
• "More on Rotations as Spin Matrix Polynomials" (Curtright, 2015)
• "Cayley transforms of su(2) representations" (Kortryk, 2015)
• "Nonsingular Efficient Modeling of Rotations in 3-space using three components" (Goldstein, 2010)
• "Cayley parametrization and the rotation group over a non-archimedean pythagorean field" (Mahmoudi, 2016)
• "Vectorial Parameterizations of Pose" (Barfoot et al., 2021)
• "Modified Gibbs's representation of rotation matrix" (Kruglov et al., 2017)