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Rodrigues' rotation formula

Rodrigues' rotation formula is a vector-based mathematical expression for performing a rotation in three-dimensional Euclidean space, transforming a given vector \mathbf{v} around a unit axis vector \mathbf{k} by an angle \theta, yielding the rotated vector \mathbf{v}' = \mathbf{v} \cos \theta + (\mathbf{k} \times \mathbf{v}) \sin \theta + \mathbf{k} (\mathbf{k} \cdot \mathbf{v}) (1 - \cos \theta).^[1] This formula decomposes the original vector into components parallel and perpendicular to the rotation axis, rotates the perpendicular component by \theta, and leaves the parallel component unchanged, ensuring the transformation preserves vector lengths and orientations in a right-handed coordinate system.^[2] Named after the French mathematician Olinde Rodrigues (1795–1851),^[3] the formula was first derived and published in his 1840 paper "Des lois géométriques qui régissent les déplacements d'un système solide dans l'espace, et de la variation des coordonnées provenant de ces déplacements considérées indépendamment des causes qui peuvent les produire", where he explored the geometric laws governing rigid body displacements and introduced parameters now known as Rodrigues vectors for composing finite rotations.^[4] Although initially overlooked, the formula gained prominence in the late 19th and 20th centuries, particularly after its connections to quaternions—independently developed by William Rowan Hamilton in 1843—were recognized, providing an alternative algebraic representation for the same rotations.^[1] In matrix form, the rotation can be expressed as a 3×3 orthogonal matrix R with determinant 1, given by R = I \cos \theta + (1 - \cos \theta) \mathbf{k} \mathbf{k}^T + \sin \theta [\mathbf{k}]_\times, where I is the identity matrix and [\mathbf{k}]_\times is the skew-symmetric cross-product matrix for \mathbf{k}, enabling efficient computation of rotated points via matrix-vector multiplication.^[2] This representation is computationally advantageous over Euler angle decompositions, as it avoids singularities like gimbal lock and directly parameterizes rotations with three components (axis direction and angle magnitude).^[1] The formula finds widespread applications in fields requiring precise 3D transformations, including computer graphics for rendering rotated objects, robotics for manipulator kinematics and path planning, aerospace engineering for spacecraft attitude control, and materials science for analyzing crystal misorientations via electron backscatter diffraction.^[5]^[6] Its efficiency and geometric intuition also make it valuable in numerical simulations of rigid body dynamics and in deriving related tools, such as the inverse Rodrigues formula for extracting axis-angle from a rotation matrix.^[7]

Core Concepts

Statement of the formula

Rodrigues' rotation formula provides an explicit expression for the result of rotating a vector in three-dimensional Euclidean space around a fixed axis by a specified angle. Given a vector \mathbf{v} \in \mathbb{R}^3 to be rotated, a unit vector \mathbf{k} \in \mathbb{R}^3 defining the axis of rotation (with \|\mathbf{k}\| = 1), and an angle \theta measured in radians, the rotated vector \mathbf{v}' is given by

\mathbf{v}' = \mathbf{v} \cos \theta + (\mathbf{k} \times \mathbf{v}) \sin \theta + \mathbf{k} (\mathbf{k} \cdot \mathbf{v}) (1 - \cos \theta).

This formula applies to proper rotations in the special orthogonal group SO(3), preserving orientation and the magnitude of the vector. The notation assumes standard vector operations: the dot product \mathbf{k} \cdot \mathbf{v} projects \mathbf{v} onto the axis, the cross product \mathbf{k} \times \mathbf{v} yields a vector perpendicular to both \mathbf{k} and \mathbf{v}, and \theta is taken in the right-handed sense with respect to \mathbf{k}. The formula holds for any \theta, including multiples of $2\pi, though rotations by \theta and \theta + 2\pi n (for integer n) yield identical results due to the periodicity of trigonometric functions. An equivalent formulation can be expressed in terms of the plane of rotation, where \mathbf{u} is a unit vector normal to the rotation plane (playing the role of \mathbf{k}): \mathbf{v}' = \cos \theta \, \mathbf{v} + \sin \theta \, (\mathbf{u} \times \mathbf{v}) + (1 - \cos \theta) (\mathbf{u} \cdot \mathbf{v}) \mathbf{u}. However, the axis-angle form is the primary and most commonly used representation for Rodrigues' formula. To illustrate, consider a two-dimensional rotation in the xy-plane, equivalent to a three-dimensional rotation around the z-axis with \mathbf{k} = (0, 0, 1) and \mathbf{v} = (x, y, 0). Substituting into the formula yields \mathbf{v}' = (x \cos \theta - y \sin \theta, x \sin \theta + y \cos \theta, 0), recovering the standard 2D rotation matrix applied to (x, y). This example highlights how the formula generalizes planar rotations by treating the out-of-plane axis appropriately.

Geometric interpretation

The Rodrigues' rotation formula provides a geometric interpretation of 3D rotations by decomposing an arbitrary vector \mathbf{v} into two components relative to the rotation axis defined by a unit vector \mathbf{k}: a parallel component \mathbf{k} (\mathbf{k} \cdot \mathbf{v}), which lies along the axis and remains unchanged during the rotation, and a perpendicular component \mathbf{v} - \mathbf{k} (\mathbf{k} \cdot \mathbf{v}), which resides in the plane orthogonal to \mathbf{k} and undergoes circular motion around the axis.^[8]^[6] This decomposition reflects Euler's rotation theorem, which states that any 3D rotation is equivalent to a single rotation about a fixed axis, preserving distances and orientations in the perpendicular plane while fixing points on the axis.^[9] Intuitively, the formula's terms capture the transformation of the perpendicular component as follows: the \cos \theta factor scales the original perpendicular vector, representing its projection onto its rotated position in the plane; the \sin \theta \, (\mathbf{k} \times \mathbf{v}) term introduces the orthogonal shift perpendicular to both the axis and the original vector, akin to a tangential displacement along the circular path via the cross product, which defines the rotation's direction; and the (1 - \cos \theta) \, \mathbf{k} (\mathbf{k} \cdot \mathbf{v}) term (or equivalently involving the double cross product \mathbf{k} \times (\mathbf{k} \times \mathbf{v})) adjusts the contribution by projecting the parallel component to account for the arc-like bending of the perpendicular part toward the axis.^[8]^[6] Visually, this manifests as a blend of scaling (via \cos \theta), shearing (through the cross product term for directional sweep), and projection (adjusting alignments in the plane), where the rotation axis \mathbf{k} scaled by the angle \theta forms the "Rodrigues vector" \boldsymbol{\theta} = \theta \mathbf{k}, encoding the finite rotation's magnitude and direction in a single vector.^[9]^[10] Special cases highlight the formula's behavior: for \theta = 0, the rotation is the identity, leaving \mathbf{v} unchanged as all trigonometric terms vanish; for \theta = \pi (a 180° rotation), the perpendicular component inverts relative to the axis, simplifying to a reflection-like operation across the plane normal to \mathbf{k}; and for small \theta, the formula approximates an infinitesimal rotation, where \sin \theta \approx \theta and $1 - \cos \theta \approx \theta^2 / 2, reducing to a linear skew-symmetric transformation dominated by the cross product term.^[8]^[6]

Derivation

Vector decomposition approach

The vector decomposition approach derives Rodrigues' rotation formula by breaking down an arbitrary vector \mathbf{v} into components parallel and perpendicular to the unit rotation axis \mathbf{k}, then applying the rotation solely to the perpendicular component while leaving the parallel component unchanged.^[8] This method leverages the geometric fact that rotation around \mathbf{k} preserves distances along the axis and rotates the orthogonal plane by the angle \theta.^[11] Begin by decomposing \mathbf{v} as \mathbf{v} = \mathbf{v}_\parallel + \mathbf{v}_\perp, where the parallel component is \mathbf{v}_\parallel = (\mathbf{k} \cdot \mathbf{v}) \mathbf{k} and the perpendicular component is \mathbf{v}_\perp = \mathbf{v} - \mathbf{v}_\parallel.^[8] Under rotation by angle \theta around \mathbf{k}, \mathbf{v}_\parallel remains invariant because it aligns with the axis.^[11] The rotated vector \mathbf{v}' thus satisfies \mathbf{v}' = \mathbf{v}_\parallel + \mathbf{v}_\perp', where \mathbf{v}_\perp' is the image of \mathbf{v}_\perp after rotation by \theta in the plane perpendicular to \mathbf{k}.^[12] To find \mathbf{v}_\perp', consider the plane spanned by \mathbf{v}_\perp and \mathbf{k} \times \mathbf{v}_\perp, which forms an orthonormal basis for the perpendicular subspace (noting that \|\mathbf{k} \times \mathbf{v}_\perp\| = \|\mathbf{v}_\perp\| since \mathbf{k} is unit).^[12] In this 2D plane, the rotation can be represented by the standard rotation matrix:

\begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix},

applied to the coordinates of \mathbf{v}_\perp relative to the basis \{\mathbf{v}_\perp / \|\mathbf{v}_\perp\|, (\mathbf{k} \times \mathbf{v}_\perp) / \|\mathbf{v}_\perp\|\}.^[8] Assuming \mathbf{v}_\perp has coordinates (r, 0) in this basis (where r = \|\mathbf{v}_\perp\|), its rotated coordinates become (r \cos \theta, r \sin \theta), yielding \mathbf{v}_\perp' = \cos \theta \, \mathbf{v}_\perp + \sin \theta \, (\mathbf{k} \times \mathbf{v}_\perp).^[12] This confirms the form of the perpendicular rotation, as the matrix rotates the vector by \theta while preserving the plane's geometry.^[11] Since \mathbf{k} \times \mathbf{v}_\perp = \mathbf{k} \times \mathbf{v} (because \mathbf{k} \times \mathbf{v}_\parallel = \mathbf{0}), substitute to obtain \mathbf{v}' = \mathbf{v}_\parallel + \cos \theta \, (\mathbf{v} - \mathbf{v}_\parallel) + \sin \theta \, (\mathbf{k} \times \mathbf{v}).^[8] Simplifying algebraically gives \mathbf{v}' = \cos \theta \, \mathbf{v} + (1 - \cos \theta) \mathbf{v}_\parallel + \sin \theta \, (\mathbf{k} \times \mathbf{v}).^[12] Inserting \mathbf{v}_\parallel = (\mathbf{k} \cdot \mathbf{v}) \mathbf{k} yields the standard Rodrigues' formula:

\mathbf{v}' = \mathbf{v} \cos \theta + (\mathbf{k} \times \mathbf{v}) \sin \theta + \mathbf{k} (\mathbf{k} \cdot \mathbf{v}) (1 - \cos \theta).

^[11] This derivation holds for any \mathbf{v} and unit \mathbf{k}, providing an explicit vector expression for the rotation.^[8]

Infinitesimal rotation method

The infinitesimal rotation method derives Rodrigues' rotation formula by modeling the continuous rotation of a vector as the solution to a linear differential equation, integrating infinitesimal changes over a finite angle. Consider a vector \mathbf{v} subject to an infinitesimal rotation by angle d\theta around a unit vector \mathbf{k}. The resulting infinitesimal change is d\mathbf{v} = \mathbf{k} \times \mathbf{v} \, d\theta, which follows from the instantaneous angular velocity \boldsymbol{\omega} = \mathbf{k} \, d\theta / dt inducing a velocity \dot{\mathbf{v}} = \boldsymbol{\omega} \times \mathbf{v}.^[13] To obtain the finite rotation, introduce the rotation matrix R(t) that transforms the initial vector to its position at time t. This matrix evolves according to the differential equation \dot{R} = [\boldsymbol{\omega}]_\times R, where [\boldsymbol{\omega}]_\times denotes the skew-symmetric cross-product matrix associated with \boldsymbol{\omega}, satisfying [\boldsymbol{\omega}]_\times \mathbf{x} = \boldsymbol{\omega} \times \mathbf{x} for any vector \mathbf{x}, with initial condition R(0) = I.^[11] The unique solution to this equation is the matrix exponential R(t) = \exp([\boldsymbol{\omega}]_\times t).^[13] For a total rotation angle \theta around \mathbf{k}, parameterize with constant angular speed such that \boldsymbol{\omega} = \mathbf{k} \theta / t over interval [0, t], or equivalently, let K = [\mathbf{k}]_\times and evaluate R(\theta) = \exp(\theta K).^[11] Expand the exponential using its Taylor series:

\exp(\theta K) = I + \theta K + \frac{\theta^2}{2!} K^2 + \frac{\theta^3}{3!} K^3 + \frac{\theta^4}{4!} K^4 + \cdots.

^[13] Since \|\mathbf{k}\| = 1, the matrix K obeys the identities K^2 = \mathbf{k} \mathbf{k}^T - I and K^3 = -K (with higher powers cycling accordingly). Substituting these relations groups the even-powered terms into a multiple of I and \mathbf{k} \mathbf{k}^T, and the odd-powered terms into a multiple of K, yielding the closed-form Rodrigues' matrix formula:

\exp(\theta K) = \cos \theta \, I + (1 - \cos \theta) \, \mathbf{k} \mathbf{k}^T + \sin \theta \, K.

^[11] Applying this rotation matrix to an arbitrary vector \mathbf{v} produces the vector form of Rodrigues' formula, where the rotated vector is a linear combination of the parallel and perpendicular components to \mathbf{k}.^[13]

Matrix Representation

Explicit rotation matrix

The Rodrigues' rotation formula can be expressed in matrix form to represent the linear transformation that rotates any vector \mathbf{v} around a unit axis \mathbf{k} = (k_x, k_y, k_z) by an angle \theta. This yields R \mathbf{v} = \cos \theta \, \mathbf{v} + \sin \theta \, (\mathbf{k} \times \mathbf{v}) + (1 - \cos \theta) (\mathbf{k} \cdot \mathbf{v}) \mathbf{k}, where R is the 3×3 rotation matrix.^[14] Substituting the cross product with its matrix equivalent gives the closed-form expression

R = \cos \theta \, I + \sin \theta \, K + (1 - \cos \theta) \, \mathbf{k} \mathbf{k}^T,

with I the identity matrix and K the skew-symmetric matrix associated with \mathbf{k},

K = \begin{pmatrix} 0 & -k_z & k_y \\ k_z & 0 & -k_x \\ -k_y & k_x & 0 \end{pmatrix}. $$ The matrix $K$ satisfies $K^T = -K$ (skew-symmetry), $K^2 = \mathbf{k} \mathbf{k}^T - I$, and $K^3 = -K$, which facilitate the derivation and computation of $R$. The explicit components of $R$ for $\mathbf{k} = (k_x, k_y, k_z)$ are

R = \begin{pmatrix} \cos \theta + k_x^2 (1 - \cos \theta) & k_x k_y (1 - \cos \theta) - k_z \sin \theta & k_x k_z (1 - \cos \theta) + k_y \sin \theta \ k_y k_x (1 - \cos \theta) + k_z \sin \theta & \cos \theta + k_y^2 (1 - \cos \theta) & k_y k_z (1 - \cos \theta) - k_x \sin \theta \ k_z k_x (1 - \cos \theta) - k_y \sin \theta & k_z k_y (1 - \cos \theta) + k_x \sin \theta & \cos \theta + k_z^2 (1 - \cos \theta) \end{pmatrix}.

[](https://mathworld.wolfram.com/RodriguesRotationFormula.html) These entries ensure $R$ belongs to the special orthogonal group SO(3), preserving lengths and orientations. Verification confirms the key properties of $R$: it fixes the rotation axis since $R \mathbf{k} = \mathbf{k}$, as the formula decomposes $\mathbf{v}$ into components [parallel](/page/Parallel) and [perpendicular](/page/Perpendicular) to $\mathbf{k}$, rotating only the latter. The [determinant](/page/Determinant) is $\det R = 1$, reflecting a proper [rotation](/page/Rotation) without [reflection](/page/Reflection).[](https://mathworld.wolfram.com/RodriguesRotationFormula.html) [Orthogonality](/page/Orthogonality) holds as $R^T R = I$, ensuring $R$ is a [rigid transformation](/page/Rigid_transformation). For a numerical example, consider [rotation](/page/Rotation) by $\theta = 90^\circ$ ($\cos \theta = 0$, $\sin \theta = 1$) around $\mathbf{k} = (0, 0, 1)$. The matrix simplifies to

R = \begin{pmatrix} 0 & -1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 1 \end{pmatrix}.

Applying $R$ to the basis vector $\mathbf{e}_1 = (1, 0, 0)$ yields $(0, 1, 0)$, consistent with a counterclockwise [rotation](/page/Rotation) in the xy-plane.[](https://mathworld.wolfram.com/RodriguesRotationFormula.html) ### Exponential map formulation The Lie algebra $\mathfrak{so}(3)$ associated with the special orthogonal group $SO(3)$ consists of all $3 \times 3$ real skew-symmetric matrices, which can be represented as $\hat{\mathbf{k}} = \begin{pmatrix} 0 & -k_z & k_y \\ k_z & 0 & -k_x \\ -k_y & k_x & 0 \end{pmatrix}$ for a vector $\mathbf{k} = (k_x, k_y, k_z)^\top \in \mathbb{R}^3$, satisfying $\hat{\mathbf{k}}^\top = -\hat{\mathbf{k}}$ and encoding [infinitesimal](/page/Infinitesimal) rotations via the cross-product operation $\hat{\mathbf{k}} \mathbf{v} = \mathbf{k} \times \mathbf{v}$ for any $\mathbf{v} \in \mathbb{R}^3$.[](https://www.cis.upenn.edu/~cis6100/geombchap14.pdf) For a [rotation](/page/Rotation) by [angle](/page/Angle) $\theta$ around the unit axis $\mathbf{k}$ (with $\|\mathbf{k}\| = 1$), the element $\theta \hat{\mathbf{k}}$ in $\mathfrak{so}(3)$ generates the corresponding finite rotation in $SO(3)$ through the [exponential map](/page/Exponential_map).[](https://www.ethaneade.com/lie.pdf) The exponential map $\exp: \mathfrak{so}(3) \to SO(3)$ is defined by the matrix exponential $\exp(\theta \hat{\mathbf{k}}) = \sum_{n=0}^\infty \frac{(\theta \hat{\mathbf{k}})^n}{n!}$, which converges to an [orthogonal matrix](/page/Orthogonal_matrix) with [determinant](/page/Determinant) 1, parameterizing all proper rotations.[](https://www.cis.upenn.edu/~cis6100/geombchap14.pdf) Since $\hat{\mathbf{k}}^2 = \mathbf{k} \mathbf{k}^\top - I$ and $\hat{\mathbf{k}}^3 = - \hat{\mathbf{k}}$ for unit $\mathbf{k}$, the powers cycle, yielding the closed-form expression:

\exp(\theta \hat{\mathbf{k}}) = \cos \theta , I + \sin \theta , \hat{\mathbf{k}} + (1 - \cos \theta) , \mathbf{k} \mathbf{k}^\top.

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