---
title: Cayley Parameterization of Rotations
url: https://www.emergentmind.com/topics/cayley-parameterization-of-rotations
type: topic
---

# Cayley Parameterization of Rotations

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., $R \in \mathrm{SO}(n)$ [2303.11835][1810.02881].

The map $\mathrm{Cay}: \mathfrak{so}(n) \to \mathrm{SO}(n)$ is bijective onto the subset of $\mathrm{SO}(n)$ consisting of those rotations for which $-1$ is not an eigenvalue (i.e., those not corresponding to a rotation by angle $\pi$ in any invariant plane). The inverse Cayley transform is
$$
A = (\!I - R)(I + R)^{-1}
$$
again defined when $(I + R)$ is invertible, i.e., for $R$ with no $-1$ eigenvalues [1612.07142][1607.07055].

The transform is real-analytic, rational in the entries of $A$, and infinitely differentiable on its domain [2303.11835].

## 2. Domain, Surjectivity, and Topological Aspects

### Domain and Range

- **Domain:** All $A \in \mathfrak{so}(n)$ such that $\det(I + A)\neq0$, equivalently, $\text{Spec}(A) \cap \{-1\} = \emptyset$.
- **Range:** All rotations $R \in \mathrm{SO}(n)$ with $\text{Spec}(R) \cap \{-1\} = \emptyset$. Rotations by precisely angle $\pi$ (in any invariant two-plane) are excluded [2303.11835][1810.02881][1607.07055].

### Local Behavior and Topology

Cayley transform is a real-analytic diffeomorphism between the domain in $\mathfrak{so}(n)$ and the open subset of SO(n) avoiding $-1$ eigenvalues. This exclusion is a measure-zero subset from the perspective of the Haar measure and the smooth structure [1810.02881].

## 3. Explicit Formulas in Low Dimensions

### SO(2)

For $A = \begin{pmatrix}0 & -a \\ a & 0\end{pmatrix}$, the Cayley transform yields
$$
R = \frac{1}{1 + a^2}
\begin{pmatrix}
1 - a^2 & -2a \\
2a & 1 - a^2
\end{pmatrix}
= 
\begin{pmatrix}
\cos\theta & -\sin\theta \\
\sin\theta & \cos\theta 
\end{pmatrix}
$$
where $\theta = 2\arctan a$. As $a \to \infty$, $\theta \to \pm\pi$, and the map becomes singular [2303.11835][1607.07055].

### SO(3)

A general $A = [\omega]_\times$ for $\omega\in\mathbb{R}^3$, yields
$$
R = (I - [\omega]_\times)(I + [\omega]_\times)^{-1}
$$
where $[\omega]_\times$ is the standard cross-product matrix. The resulting rotation is about axis $\hat{\omega}$ with angle $\varphi = 2\arctan \|\omega\|$. The explicit formula is
$$
R = I + \frac{2}{1+\|\omega\|^2}[\omega]_\times + \frac{2}{1+\|\omega\|^2}[\omega]_\times^2
$$
The denominator enforces the exclusion of rotations with $\varphi=\pi$ ($\|\omega\|\rightarrow\infty$) [2303.11835][1607.07055][1810.02881].

### Component Formula Table (n=3)

| Entry         | Formula (with $\|a\|^2 = u^2 + v^2 + w^2$)   |  
|--------------|-----------------------------------------------|
| $R_{11}$     | $\dfrac{1 + u^2 - v^2 - w^2}{1 + u^2 + v^2 + w^2}$ |  
| $R_{12}$     | $\dfrac{2(uv - w)}{1 + u^2 + v^2 + w^2}$      |  
| $R_{13}$     | $\dfrac{2(uw + v)}{1 + u^2 + v^2 + w^2}$      |  
| $R_{21}$     | $\dfrac{2(uv + w)}{1 + u^2 + v^2 + w^2}$      |  
| $R_{22}$     | $\dfrac{1 - u^2 + v^2 - w^2}{1 + u^2 + v^2 + w^2}$ |  
| $R_{23}$     | $\dfrac{2(vw - u)}{1 + u^2 + v^2 + w^2}$      |  
| $R_{31}$     | $\dfrac{2(uw - v)}{1 + u^2 + v^2 + w^2}$      |  
| $R_{32}$     | $\dfrac{2(vw + u)}{1 + u^2 + v^2 + w^2}$      |  
| $R_{33}$     | $\dfrac{1 - u^2 - v^2 + w^2}{1 + u^2 + v^2 + w^2}$ |  

This rational structure is a hallmark of the Cayley parameterization [1105.3828][1810.02881][1607.07055].

## 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 $X\in\mathfrak{so}(n)$ is given by $2^{n(n-1)/2}\cdot\det(I_n+X)^{-n}$, directly encoding the volume change under the map from Lie algebra to rotation group [1810.02881].

### Comparison Table

| Parameterization | Domain            | Surjectivity     | Regularity            | Notable Features                   |
|------------------|-------------------|------------------|-----------------------|------------------------------------|
| Cayley           | $\mathfrak{so}(n)$, $\det(I+A)\ne0$ | Not onto full SO(n) (misses angle $\pi$) | Rational, analytic | Simple algebraic form, no transcendentals; excludes measure-zero subset |
| Exponential      | $\mathfrak{so}(n)$ | Onto SO(n) ($n=2,3$); onto component of I ($n>3$) | Analytic, transcendental | Requires trigonometric functions; global but multi-valued [2303.11835][1703.00300] |
| Quaternions (n=3)| $q \in \mathbb{H}, \|q\|=1$ | Double cover    | Rational entries, with unit-norm constraint | Smooth, covers all SO(3); $q\equiv-q$ redundancy [2303.11835][1703.00300] |
| Euler Angles     | $\mathbb{R}^3$    | Local charts, possible singularities | Transcendental | Minimal, but can suffer “gimbal lock” [2303.11835] |

## 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 [2303.11835][1810.02881][2103.07309].
- The rational formula enables efficient computation of derivatives and Jacobians, particularly advantageous for optimization and geometric integration [2303.11835][1810.02881][1506.00500][2103.07309].
- For $n=3$, the composition of two Cayley-parameterized rotations admits a closed-form algebraic expression: for $u_1, u_2 \in \mathbb{R}^3$,
$$
u_3 = \frac{u_1 + u_2 + u_2 \times u_1}{1 - u_2 \cdot u_1}
$$
so that $R(u_3) = R(u_2) R(u_1)$, facilitating direct computations [1607.05999].

In high-dimensional settings (large $n$), the cost of inverting $(I+A)$ may become a computational bottleneck; however, for structured $A$ (e.g., sparse or low-rank), this can often be mitigated [2303.11835][1810.02881].

## 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 [2303.11835].
- 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 [1810.02881].
- The Cayley transform defines local diffeomorphisms on Stiefel and Grassmann manifolds, making it central to optimization algorithms with orthogonality constraints [1612.07142][1810.02881].
- 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 [1105.3828].

### 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): $R_{\mathrm{Cayley}}(\theta) = (1 - i t J)(1 + i t J)^{-1}$, $t = \tan(\theta/2)$, where $J$ is the angular momentum operator. This yields explicit polynomial expressions for arbitrary spin, with the structure of coefficients governed by central factorial numbers [1506.00500][1506.04648].
- 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 $\pi$-rotations (or, more generally, elements of SO(n) with $-1$ eigenvalues) is inherent: as the parameter $\|A\|$ or $\|u\|$ diverges, the associated rotation angle approaches $\pi$, and the Cayley map becomes singular [2303.11835][1607.05999][2103.07309].
- 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 [1005.4661][1810.02881].
- 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 [1703.00300][1607.05999].
- 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 [1506.00500].

## References

- "Lipschitz-bounded 1D convolutional neural networks using the Cayley transform and the controllability Gramian" [2303.11835]
- "An Algorithmic Solution to the Five-Point Pose Problem Based on the Cayley Representation of Rotations" [1105.3828]
- "Random orthogonal matrices and the Cayley transform" [1810.02881]
- "Cayley Transform on Stiefel manifolds" [1612.07142]
- "Visualizing Rotations and Composition of Rotations with Rodrigues' Vector" [1607.05999]
- "More on Rotations as Spin Matrix Polynomials" [1506.04648]
- "Cayley transforms of su(2) representations" [1506.00500]
- "Nonsingular Efficient Modeling of Rotations in 3-space using three components" [1005.4661]
- "Cayley parametrization and the rotation group over a non-archimedean pythagorean field" [1607.07055]
- "Vectorial Parameterizations of Pose" [2103.07309]
- "Modified Gibbs's representation of rotation matrix" [1703.00300]

Source: https://www.emergentmind.com/topics/cayley-parameterization-of-rotations